Sequence Numbers: Fibonacci
Type Sequences
By David Brooks
This post contains several examples of
Sequence numbers that produce the same terms as Fibonacci like sequences.
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A00045: The Fibonacci
Sequence: a(0) = 0, a(1) = 1, an when n>1 then a(n) = a(n-1) + a(n-2).
The digital expansion of
the inverse of 999,999,998,999,999,999 produces the first 43 terms of the
Fibonacci Sequence in nine digit chunks.
1/999999998999999999 =
0.
000000000 000000001 000000001 000000002 000000003 000000005 000000008 000000013 000000021 000000034 000000055 000000089 000000144 000000233 000000377 000000610 000000987 000001597 000002584 000004181 000006765 000010946 000017711 000028657 000046368 000075025 000121393 000196418 000317811 000514229 000832040 001346269 002178309 003524578 005702887 009227465 014930352 024157817 039088169 063245986 102334155 165580141 267914296 433494437 … |
The Super Fibonacci
Sequence: a(0) = 0, a(1) = 1, an when n>1 then a(n) = a(n-1) + 2*a(n-2).
The digital expansion of
the inverse of 999,999,998,999,999,998 produces the first 30 terms of the
Fibonacci Sequence in nine digit chunks.
1/999999998999999998 =
0.
000000000 000000001 000000001 000000003 000000005 000000011 000000021 000000043 000000085 000000171 000000341 000000683 000001365 000002731 000005461 000010923 000021845 000043691 000087381 000174763 000349525 000699051 001398101 002796203 005592405 011184811 022369621 044739243 089478485 178956971 357913941 … |
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A000073: The Tribonacci
Sequence. A(0) = a(1) = 0, a(2) = 1,
and when n>2 then a(n) = a(n-1) + a(n-2) + a(n-3).
999,999,998,999,999,998,999, 999,999
1/99999999899999999899999
9999 =
0.
000000000 000000000 000000001 000000001 000000002 000000004 000000007 000000013 000000024 000000044 000000081 000000149 000000274 000000504 000000927 000001705 000003136 000005768 000010609 000019513 000035890 000066012 000121415 000223317 000410744 000755476 001389537 002555757 004700770 008646064 015902591 029249425 053798080 098950096 181997601 334745777 … |
The Super-Tribonacci
Sequence: a(0) = a(1) = 0, a(2) = 1, and when n>2 then a(n) = a(n-1) +
2*a(n-2) + 3*a(n-3).
999,999,998,999,999,997,999,999,997
1/999999998999999997999999997
=
0.
000000000 000000000 000000001 000000001 000000003 000000008 000000017 000000042 000000100 000000235 000000561 000001331 000003158 000007503 000017812 000042292 000100425 000238445 000566171 001344336 003192013 007579198 017996232 042730667 101460725 240910755 … |
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The 1, -1, 1 Tribonacci
Sequence
The Sequence Number is:
999,000,999
1/999000999=
0.
000 000 001 001 000 000 001 001 000 000 001 001 000 000 001 001 000 000 001 001 000 000 001 001 000 000 001 001 000 000 001 001 …
Terms are written in
three digit strings.
Terms are accurate up to
infinity (this is a repeating decimal with a period of 12).
Compare with OEIS sequence
A021913.
This sequence can
actually be produced by a shorter Sequence Number:
909
1/909 =
0.0011001100110011001100110011 00110011001100110011001100110011
...
This is a repeating
decimal with a period of 4.
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A000078: The Tetranacci
Sequence: a(0) = a(1) = a(2) = 0, a(3) = 1, and when n>3 then a(n) =
a(n-1) + a(n–2) + a(n-3) + a(n-4).
999,999,998,999,999,998,999,999,
998,999,999,999
1/99999999899999999899999999899 9999999
=
0.
000000000 000000000 000000000 000000001.000000001 000000002 000000004 000000008 000000015 000000029 000000056 000000108 000000208 000000401 000000773 000001490 000002872 000005536 000010671 000020569 000039648 000076424 000147312 000283953 000547337 001055026 002033628 003919944 007555935 014564533 028074040 054114452 104308960 201061985 387559437 … |
The Super-Tetranacci
Sequence: a(0) = a(1) = a(2) = 0, a(3) = 1, and when n>3 then a(n) =
a(n-1) + 2*a(n–2) + 3*a(n-3) + 4*a(n-4).
999,999,998,999,999,997,999,999,
996,999,999,996
1/999999998999999997999999996 999999996
=
0.
000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000050 000000128 000000323 000000813 000002043 000005150 000012967 000032648 000082204 000207001 000521221 001312427 003304688 008321209 020952750 052758940 132846819 334507785 … |
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The 1, 0, -1, 1
Tetranacci Sequence
The Sequence Number is:
9,009
1/9009 =
0.
000111000111000111000111000111 000111000111000111000111000111...
Terms are written in one
digit strings.
Terms are accurate up to
infinity because this is a repeating fraction with a period of 6.
Compare with OEIS
sequence A115360.
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A Special 2,0,1,6
Tetranacci Sequence Number for 2016:
This is a special
Tetranacci Sequence defined as : a(0) = a(1) = a(2) = 0, a(3) = 1, and when
n>3 then a(n) = 2 * a(n-1) + 0 * a(n–2) + 1 * a(n-3) + 6 * a(n-4).
The digital expansion of
the inverse of the sequence number 9,999,999,999,999,997,999,999,
999,999,999,999,999,999,999, 999,989,999,999,999,999,994 produces a digital sequence that shows the terms of the special Tetranacci Sequence defined in the paragraph above. It will write the terms in 16 digit strings. It is accurate to the 40th non-zero term.
1/9999999999999997999999999 99999999999999999999998999
9999999999994 =
0.
0000000000000000 0000000000000000 0000000000000000 0000000000000001 0000000000000002 0000000000000004 0000000000000009 0000000000000026 0000000000000068 0000000000000169 0000000000000418 0000000000001060 0000000000002697 0000000000006826 0000000000017220 0000000000043497 0000000000110002 0000000000278180 0000000000703177 0000000001777338 0000000004492868 0000000011357993 0000000028712386 0000000072581668 0000000183478537 0000000463817418 0000001172490820 0000002963950185 0000007492589010 0000018940573348 0000047880041801 0000121036373722 0000305968854852 0000773461191593 0001955239007714 0004942665112612 0012494604545929 0031585215249130 0079844529657156 0201839654535913 0510232151596530 1289820124344996 3260547081168841 … |
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A001591: The Pentanacci
Sequence: a(0) = a(1) = a(2) = a(3) 0, a(4) = 1, and when n>4 then a(n) =
a(n-1) + a(n–2) + a(n-3) + a(n-4) + a(n -5).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,999
1/999999998999999998999999998 999999998999999999
=
0.
000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000031 000000061 000000120 000000236 000000464 000000912 000001793 000003525 000006930 000013624 000026784 000052656 000103519 000203513 000400096 000786568 001546352 003040048 005976577 011749641 023099186 045411804 089277256 175514464 345052351 … |
The Super-Pentanacci
Sequence: a(0) = a(1) = a(2) = a(3) = 0, a(4) = 1, and when n>4 then a(n)
= a(n-1) + 2*a(n–2) + 3*a(n-3) + 4*a(n-4) + 5*a(n-5).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,995
1/999999998999999997999999996 999999995999999995
=
0.
000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000138 000000358 000000923 000002378 000006125 000015772 000040638 000104684 000269666 000694661 001789457 004609703 011874684 030589435 078799045 202988064 … |
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90,909 produces the
terms of the 1,
-1, 1, -1, 1 Pentanacci sequence, using single digit strings. The 1,-1,1,-1,1 Pentanacci Sequence: defined as a(0) = a(1) = a(2) = a(3) = 0, a(4) = 1, and when n>4 then a(n) = a(n-1) – a(n-2) + a(n-3) – a(n-4) + a(n+4).
1/90909 = 0. 0 0 0 0 1 1
0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0…
This is a repeating
decimal with a period of six.
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90,009 produces the 1,
0, 0, -1, 1 Pentanacci sequence (defined as: a(0) = a(1) = a(2) = a(3) = 0,
a(1) = 1, and when n>4 then a(n) = a(n-1) – a(n-2) – a(n-4) + a(n-5).),
and written in single digit strings.
1/90009 = 0. 0 0 0 0 1 1
1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 ...
This is a repeating
decimal with a period of eight.
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A001592: The Hexanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = 0, a(5) = 1, and when n>5
then a(n) = a(n-1) + a(n–2) + a(n-3) + a(n-4) + a(n -5) + a(n-6).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,999
1/999999998999999998999999998 999999998999999998999999999
=
0.
000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000063 000000125 000000248 000000492 000000976 000001936 000003840 000007617 000015109 000029970 000059448 000117920 000233904 000463968 000920319 001825529 003621088 007182728 014247536 028261168 056058368 111196417 220567305 437513522 … |
The Super-Hexanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = 0, a(5) = 1, and when n>5
then a(n) = a(n-1) + 2*a(n–2) + 3*a(n-3) + 4*a(n-4) + 5*a(n-5) + 6*a(n-6).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,994
1/999999998999999997 9999999969999999959 99999994999999994
=
0.
000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000370 000000965 000002510 000006527 000016972 000044130 000114740 000298364 000775817 002017307 005245475 013639476 035465875 092219749 239793264 … |
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A122189: The Heptanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 1, and when
n>6 then a(n) = a(n-1) + a(n–2) + a(n-3) + a(n-4) + a(n -5) + a(n-6) +
a(n-7).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,999
1/999999998999999998 999999998999999998 999999998999999998
999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000127 000000253 000000504 000001004 000002000 000003984 000007936 000015808 000031489 000062725 000124946 000248888 000495776 000987568 001967200 003918592 007805695 015548665 030972384 061695880 122895984 244804400 487641600 … |
The Super-Heptanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 1, a(6) = 1, and when
n>6 then a(n) = a(n-1) + 2*a(n–2) + 3*a(n-3) + 4*a(n-4) + 5*a(n-5) +
6*a(n-6) + 7*a(n-7).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,993
1/999999998999999997 999999996999999995 999999994999999993
999999993 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000979 000002559 000006681 000017441 000045530 000118856 000310271 000809949 002114383 005519581 014408825 037614133 098191421 256327992 … |
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“Beechwood 4-5789"
is a song written by Marvin Gaye, William "Mickey" Stevenson and
George Gordy. It was a 1962 hit single
for the Motown girl group “The Marvelettes”.
The title of the song refers to a phone number – 234-5789. If you want to listen to this song and see
a picture of the Marvelettes you can visit this YouTube website: https://www.youtube.com/watch?v=Us18AUBM2RI
We can create a
“Beechwood 4-5789" Heptanacci Sequence and it’s associated Sequence
Number, based on this famous phone number.
The “Beechwood 4-5789"
Heptanacci Sequence:
a(0) = a(1) = a(2) =
a(3) = a(4) = a(5) = 0, a(6) = 1, and when n>6 then a(n) = 2*a(n-1) +
3*a(n-2) + 4*a(n-3) + 5*a(n-4) + 7*a(n-5) + 8*a(n-6) + 9*a(n-7).
The Sequence Number is:
9,999,999,999,997,999,999,999, 999,699,999,999,999,959,999,999, 999,994,999,999,999,999,299,999, 999,999,919,999,999,999,991
1/9999999999997999999999999
699999999999959999999999994 999999999999299999999999919 999999999991 =
0.
0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000001 0000000000002 0000000000007 0000000000024 0000000000082 0000000000281 0000000000961 0000000003287 0000000011233 0000000038405 0000000131301 0000000448897 0000001534708 0000005246912 0000017938323 0000061328143 0000209670775 0000716829672 0002450721977 0008378612714 0028645089715 0097932819316 0334816095709 …
Terms are written in 13
digit strings so that all terms up to 12 digits long will be accurate.
Terms are accurate up to
the 23rd non-zero term.
This sequence is not
listed in the OEIS – yet.
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"867-5309/Jenny"
is a song written by Alex Call and Jim Keller and performed by Tommy Tutone
that was released on the album Tommy Tutone 2, on the Columbia Records label.
It peaked at #4 on the Billboard Hot 100 chart and #16 on the Billboard Top
Tracks chart in 1982. If you want to
see a video of Tommy Tutone singing this song you can visit the YouTube
website: https://www.youtube.com/watch?v=6WTdTwcmxyo
This song actually
started a nation-wide disturbance because of the number of people calling
this number and asking for Jenny. See
the Wikipedia article: https://en.wikipedia.org/wiki/867-5309/Jenny
We can create a
"867-5309/Jenny" Heptanacci Sequence and it’s associated Sequence
Number, based on this famous phone number.
The
"867-5309/Jenny" Heptanacci Sequence:
a(0) = a(1) = a(2) =
a(3) = a(4) = a(5) = 0, a(6) = 1, and when n>6 then a(n) = 8*a(n-1) +
6*a(n-2) + 7*a(n-3) + 5*a(n-4) + 3*a(n-5) + 0*a(n-6) + 9*a(n-7).
The Sequence Number is:
9,999,999,999,991,999,999,999, 999,399,999,999,999,929,999, 999,999,994,999,999,999,999, 699,999,999,999,999,999,999, 999,991
1/9999999999991999999999999
399999999999929999999999994 999999999999699999999999999 999999999991 =
0.
0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000000 0000000000001 0000000000008 0000000000070 0000000000615 0000000005401 0000000047431 0000000416533 0000003657951 0000032123745 0000282107385 0002477437700 0021756600088 0191064198056 …
Terms are written in 13
digit strings so that all terms up to 12 digits long will be accurate.
Terms are accurate up to
the 13th non-zero term.
This sequence is not
listed in the OEIS – yet.
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A079262: The Octanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 0, a(7) = 1 and
when n>7 then a(n) = a(n-1) + a(n–2) + a(n-3) + a(n-4) + a(n -5) + a(n-6)
+ a(n-7) + a(n-8).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,999
1/999999998999999998 999999998999999998 999999998999999998 999999998999999999
=
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000255 000000509 000001016 000002028 000004048 000008080 000016128 000032192 000064256 000128257 000256005 000510994 001019960 002035872 004063664 008111200 016190208 032316160 064504063 128752121 256993248 …. |
The Super-Octanacci Sequence:
a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 0, a(7) = 1 and when n>7
then a(n) = a(n-1) + 2*a(n–2) + 3*a(n-3) + 4*a(n-4) + 5*a(n-5) + 6*a(n-6) +
7*a(n-7) + 8*a(n-8).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,992
1/999999998999999997 999999996999999995 999999994999999993999999992999999992
=
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002575 000006737 000017617 000046066 000120456 000314975 000823613 002153623 005631389 014725265 038504373 100683197 263271560 … |
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10/01/2015
The 1, 0, 0, 1, 2, 0, 1,
5 Octanacci Sequence a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 0, a(7)
= 1, and when n>7 then a(n) = a(n-1) + a(n-4) + 2*a(n-5) + a(n-7) +
5*a(n-8).
The Sequence Number is:
999,998, 999,999,999,999,999,998,
999,997,999,999,999,998, 999,995
1/999998999999999999999998 999997999999999998999995
=
0.
000000 000000 000000 000000 000000 000000 000000 000001 000001 000001 000001 000002 000005 000008 000012 000022 000037 000061 000096 000157 000271 000458 000758 001254 002085 003486 005797 009623 016029 026733 044546 074118 123304 205322 341942 569296 …
Terms are written in six
digit strings.
Terms are accurate up to
569,296, which is the 29th non-zero term.
This sequence is not
listed in the OEIS.
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A104144: The Nonanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 0, a(8) = 1
and when n>8 then a(n) = a(n-1) + a(n–2) + a(n-3) + a(n-4) + a(n -5) +
a(n-6) + a(n-7) + a(n-8) + a(n-9).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000511 000001021 000002040 000004076 000008144 000016272 000032512 000064960 000129792 000259328 000518145 001035269 002068498 004132920 008257696 016499120 032965728 065866496 131603200 262947072 … |
The Super-Nonanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 0, a(8) =
1, and when n>8 then a(n) = a(n-1) + 2*a(n–2) + 3*a(n-3) + 4*a(n-4) +
5*a(n-5) + 6*a(n-6) + 7*a(n-7) + 8*a(n-8) + 9*a(n-9).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,991
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999991 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006755 000017680 000046264 000121059 000316775 000828905 002168995 005675606 014851346 038861475 101688785 266088875 … |
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A122265: The Decanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 0,
a(9) = 1 and when n>9 then a(n) = a(n-1) + a(n–2) + a(n-3) + a(n-4) + a(n
-5) + a(n-6) + a(n-7) + a(n-8) + a(n-9) + a(n-10).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001023 000002045 000004088 000008172 000016336 000032656 000065280 000130496 000260864 000521472 001042432 002083841 004165637 008327186 016646200 033276064 066519472 132973664 265816832 … |
The Super-Decanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 0,
a(9) = 1, and when n>9 then a(n) = a(n-1) + 2*a(n–2) + 3*a(n-3) + 4*a(n-4)
+ 5*a(n-5) + 6*a(n-6) + 7*a(n-7) + 8*a(n-8) + 9*a(n-9) + 10*a(n-10).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,990
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999990 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017700 000046334 000121279 000317445 000830905 002174875 005692686 014900476 039001655 102085935 267207575 … |
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A168082: The Undecanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9)
= 0, a(10) = 1 and when n>10 then a(n) = a(n-1) + a(n–2) + a(n-3) + a(n-4)
+ a(n -5) + a(n-6) + a(n-7) + a(n-8) + a(n-9) + a(n-10) + a(n-11).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999,
999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002047 000004093 000008184 000016364 000032720 000065424 000130816 000261568 000523008 001045760 002091008 004180992 008359937 016715781 033423378 066830392 133628064 267190704 … |
The Super-Undecanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9)
= 0, a(10) = 1, and when n>10 then a(n) = a(n-1) + 2*a(n–2) + 3*a(n-3) +
4*a(n-4) + 5*a(n-5) + 6*a(n-6) + 7*a(n-7) + 8*a(n-8) + 9*a(n-9) + 10*a(n-10)
+ 11*a(n-11).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999,
989
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999989 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046356 000121356 000317687 000831642 002177075 005699154 014919264 039055698 102240133 267644550 … |
|
A168083: The Dodecanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9)
= a(10) = 0, a(11) = 1 and when n>11 then a(n) = a(n-1) + a(n–2) + a(n-3)
+ a(n-4) + a(n -5) + a(n-6) + a(n-7) + a(n-8) + a(n-9) + a(n-10) + a(n-11) +
a(n-12).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999998999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002048 000004095 000008189 000016376 000032748 000065488 000130960 000261888 000523712 001047296 002094336 004188160 008375296 016748544 033492993 066977797 133939218 267845688 … |
The Super-Dodecanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9)
= a(10)= 0, a(11) = 1, and when n>11 then a(n) = a(n-1) + 2*a(n–2) +
3*a(n-3) + 4*a(n-4) + 5*a(n-5) + 6*a(n-6) + 7*a(n-7) + 8*a(n-8) + 9*a(n-9) +
10*a(n-10) + 11*a(n-11) + 12*a(n-12).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999, 988,999,999,988
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999988999999988 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046368 000121380 000317771 000831906 002177879 005701554 014926320 039076194 102299089 267812766 … |
|
A168084: The Tridecanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9)
= a(10) = a(11) = 0, a(12) = 1 and when n>12 then a(n) = a(n-1) + a(n–2) +
a(n-3) + a(n-4) + a(n -5) + a(n-6) + a(n-7) + a(n-8) + a(n-9) + a(n-10) +
a(n-11) + a(n-12) + a(n-13).
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,998,999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001.000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002048 000004096 000008191 000016381 000032760 000065516 000131024 000262032 000524032 001048000 002095872 004191488 008382464 016763904 033525760 067047424 134086657 268156933 … |
The Super Tridecanacci
Sequence: This is a Fibonacci like
sequence where a(0) = a(1) = a(2) = a(3) = … = a(11) = 0, a(12) = 1, and when
n>12 then a(n) = a(n – 1) + 2*a(n – 2) + 3*a(n – 3) + 4*a(n – 4) + 5*a(n –
5) + 6*a(n – 6) + 7*a(n – 7) + 8*a(n – 8) + 9*a(n – 9) + 10*a(n – 10) +
11*a(n – 11) + 12*a(n – 12) + 13*a(n – 13).
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999, 988,999,999,987,999,999,987
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999988999999987 999999987 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046368 000121393 000317797 000831997 002178165 005702425 014928920 039083838 102321293 267876635 … |
|
The Tetradecanacci
Sequence: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9)
= a(10) = a(11) = a(12) = 0, a(13) = 1 and when n>13 then a(n) = a(n-1) +
a(n–2) + a(n-3) + a(n-4) + a(n -5) + a(n-6) + a(n-7) + a(n-8) + a(n-9) +
a(n-10) + a(n-11) + a(n-12) + a(n-13) + a(n-14).
999,998,999,998,999,998,999,998,
999,998,999,998,999,998,999,998, 999,998,999,998,999,998,999,998, 999,998,999,999
1/999998999998999998999998
999998999998999998999998 999998999998999998999998 999998999999 =
0.
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000001 000001 000002 000004 000008 000016 000032 000064 000128 000256 000512 001024 002048 004096 008192 016383 032765 065528 131052 262096 … |
The Super Tetradecanacci
Sequence: This is a Fibonacci like
sequence where a(0) = a(1) = a(2) = a(3) = … = a(11) = a(12) = 0, a(13) = 1,
and when n>13 then a(n) = a(n – 1) + 2*a(n – 2) + 3*a(n – 3) + 4*a(n – 4)
+ 5*a(n – 5) + 6*a(n – 6) + 7*a(n – 7) + 8*a(n – 8) + 9*a(n – 9) + 10*a(n –
10) + 11*a(n – 11) + 12*a(n – 12) + 13*a(n – 13) + 14*a(n-14).
999,998,999,997,999,996,999,995,
999,994,999,993,999,992,999,991, 999,990,999,989,999,988,999,987, 999,986,999,986
1/99999899999799999699999599999
99999399999299999199999099998 9999988999987999986999986 =
0.
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000001 000001 000003 000008 000021 000055 000144 000377 000987 002584 006765 017711 046368 121393 317811 … |
|
A220493: Pentadecanacci
Sequence
This is a Fibonacci like sequence defined as: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = a(11) = a(12) = a(13) = 0, a(14) = 1 and when n>14 then a(n) = a(n-1) + a(n–2) + a(n-3) + a(n-4) + a(n -5) + a(n-6) + a(n-7) + a(n-8) + a(n-9) + a(n-10) + a(n-11) + a(n-12) + a(n-13) + a(n-14) + a(n-15).
999,998,999,998,999,998,999,998,
999,998,999,998,999,998,999,998, 999,998,999,998,999,998,999,998, 999,998,999,998,999,999
1/999998999998999998999998
999998999998999998999998 999998999998999998999998 999998999998999999 =
0.
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000001 000001 000002 000004 000008 000016 000032 000064 000128 000256 000512 001024 002048 004096 008192 016384 032767 065533 131064 262124 …. |
Super-Pentadecanacci
Sequence
This is a Fibonacci like sequence defined as: a(0) = a(1) = a(2) = a(3) = … = a(12) = a(13) = 0, a(14) = 1, and when n>14 then a(n) = a(n – 1) + 2*a(n – 2) + 3*a(n – 3) + 4*a(n – 4) + 5*a(n – 5) + 6*a(n – 6) + 7*a(n – 7) + 8*a(n – 8) + 9*a(n – 9) + 10*a(n – 10) + 11*a(n – 11) + 12*a(n – 12) + 13*a(n – 13) + 14*a(n-14) + 15*a(n-15).
999,998,999,997,999,996,999,995,
999,994,999,993,999,992,999,991, 999,990,999,989,999,988,999,987, 999,986,999,985,999,985
1/999998999997999996999995
999994999993999992999991 999990999989999988999987 999986999985999985 =
0.
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000001 000001 000003 000008 000021 000055 000144 000377 000987 002584 006765 017711 046368 121393 317811 … |
|
A249169 Hexadecanacci
Sequence
Sequence Number:
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002048 000004096 000008192 000016384 000032768 000065535 000131069 000262136 000524268 001048528 002097040 004194048 008388032 016775936 033551616 067102720 134204416 268406784 … |
Super Hexadecanacci
Sequence
Sequence Number:
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999, 988,999,999,987,999,999,986,999, 999,985,999,999,984,999,999,984
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999988999999987 999999986999999985999999984 999999984 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046368 000121393 000317811 000832040 002178309 005702870 014930300 039087995 102333597 267912567 … |
|
Heptadecanacci Sequence
Sequence Number:
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002048 000004096 000008192 000016384 000032768 000065536 000131071 000262141 000524280 001048556 002097104 004194192 008388352 016776640 033553152 067106048 134211584 268422144 … |
Super Heptadecanacci
Sequence
Sequence Number:
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999, 988,999,999,987,999,999,986,999, 999,985,999,999,984,999,999,983, 999,999,983
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999988999999987 999999986999999985999999984 999999983999999983 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046368 000121393 000317811 000832040 002178309 005702887 014930334 039088114 102333971 267913706 … |
|
Octadecanacci Sequence
Sequence Number:
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999998999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002048 000004096 000008192 000016384 000032768 000065536 000131072 000262143 000524285 001048568 002097132 004194256 008388496 016776960 033553856 067107584 134214912 268429312 … |
Super Octadecanacci
Sequence
Sequence Number:
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999, 988,999,999,987,999,999,986,999, 999,985,999,999,984,999,999,983, 999,999,982,999,999,982
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999988999999987 999999986999999985999999984 999999983999999982999999982 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046368 000121393 000317811 000832040 002178309 005702887 014930352 039088150 102334097 267914102 … |
|
Nonadecanacci Sequence
Sequence Number:
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 99999999899999999899999999 999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002048 000004096 000008192 000016384 000032768 000065536 000131072 000262144 000524287 001048573 002097144 004194284 008388560 016777104 033554176 067108288 134216448 268432640 |
Super Nonadecanacci
Sequence
Sequence Number:
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999, 988,999,999,987,999,999,986,999, 999,985,999,999,984,999,999,983, 999,999,982,999,999,981,999,999, 981
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999988999999987 999999986999999985999999984 999999983999999982999999981 999999981 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046368 000121393 000317811 000832040 002178309 005702887 014930352 039088169 102334135 267914235 … |
|
Icosadecanacci Sequence
Sequence Number:
999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,998,999,999,998,999, 999,998,999,999,998,999,999,998, 999,999,998,999,999,998,999,999, 998,999,999,999
1/999999998999999998999999998
999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999998999999998 999999998999999999 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000002 000000004 000000008 000000016 000000032 000000064 000000128 000000256 000000512 000001024 000002048 000004096 000008192 000016384 000032768 000065536 000131072 000262144 000524288 001048575 002097149 004194296 008388588 016777168 033554320 067108608 134217152 268434176 … |
Super Icosadecanacci
Sequence
Sequence Number:
999,999,998,999,999,997,999,999,
996,999,999,995,999,999,994,999, 999,993,999,999,992,999,999,991, 999,999,990,999,999,989,999,999, 988,999,999,987,999,999,986,999, 999,985,999,999,984,999,999,983, 999,999,982,999,999,981,999,999, 980,999,999,980
1/999999998999999997999999996
999999995999999994999999993 999999992999999991999999990 999999989999999988999999987 999999986999999985999999984 999999983999999982999999981 999999980999999980 =
0.
000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000000 000000001 000000001 000000003 000000008 000000021 000000055 000000144 000000377 000000987 000002584 000006765 000017711 000046368 000121393 000317811 000832040 002178309 005702887 014930352 039088169 102334155 267914275 … |
David
This can be explained in an easy way using generating functions.
ReplyDelete