Here are some calculations that I thought were interesting, and I'm not sure I posted.
I particularly like the last one.
The 1,2,3,4,5 Pentanacci sequence
0.000000000000000000000000000001000001000003
00000800002100005500013800035800092300237800 6125015772040638104684269666694661^-1
9.999989999979999969999959999950000000000000
000000000000... × 10^29
The Sequence Number is:
999,998,999,997,999,996,999,995,999,995
0.000005010015020025030035040045050055060
065070075080085090095100105110115120125130...
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Fibonacci Bisection, A001906
Every Other Fibonacci Numbers (the even
numbered terms)
First 16 terms
0, 1, 3, 8, 21, 55, 144, 377, 987,
2584, 6765, 17711, 46368, 121393, 317811, 832040
0.00000000000100000300000800002100005500014400
0377000987002584006765017711046368121393317811 832040^-1
9.999970000010000000000000000000000000000000000
0000000
00...
× 10^11
It looks like the sequence number will
be 999997000001.
So let’s check it out – see if it
works.
1/999997000001 =
0.
000000 000001 000003 000008 000021 000055 000144 000377 000987 002584 006765 017711 046368 121393 … |
The Other Every Other Fibonacci
Numbers, A001519
OEIS describes this sequence as a
bisection of the Fibonacci Sequence, listing the odd terms – but they include
both 1s in this sequence and one of these must be an even numbered term. I have elected to correct this in my data
listed below.
(The odd numbered terms from the
Fibonacci Sequence A000045)
1, 2, 5, 13, 34, 89, 233, 610, 1597,
4181, 10946, 28657, 75025,
196418, 514229,
0.000001000002000005000013000034000089000233000
610001597004181010946028657075025196418514229^-1
999997.999998999998999998999998999998999998999998
9999989999989999989999989999990000003462661394988 2168698964615760532556449352366148282944199740116 536033331950127866
9239649635840932689458071744735699145762301934204
21485260424710238770927442490...
So the Sequene Number is:
9999979999989999989999989999989999989999989999
98999998999998999998999998999999
1/999997999998999998999998999998999998999998999
998999998999998999998999998999999 =
0.
000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000001 000002 000005 000013 000034 000089 000233 000610 001597 004181 010946 028657 075025 196418
Written in six digit strings.
Accurate up to the 14th
non-zero term
Compare with OEIS sequence A001519.
Can we do better?
This looks like the
2,1,1,1,1,1,1,1,1,1,1,1,1 Tridecanacci Sequence, written to list terms in 12
digit stings: a(0) = a(1) = … = a(10) = a(11) = 0, a(12) = 1, and when
n>12 then a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) +
a(n-6) + a(n-8) + a(n-9) + a(n-10) + a(n-11) + a(n-12) + a(n-13).
I rewrote the sequence number to
produce terms in 12 digit string – sadly the terms were not accurate. Well, you can’t win them all. Maybe someone else will pick up this challenge
and find a solution.
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Triangular Numbers, A000217
Finding a Sequence Number for
Triangular Numbers:
First we start with a list of the
Triangular Number. I decided to end
the
list at 300 (no special reason, but I had to choose somewhere to stop):
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55,
66, 78, 91, 105, 120, 136, 153,
171, 190, 210, 231, 253, 276, 300
Then I convert each number to a six
digit string by adding zeros in
front of each term where needed, remove the
comas and spaces,
and put a “0.” in front and a “^-1” on the tail end.
0.00000000000100000300000600001000001500002100
0028000036000045000055000066000078000091000105 000120000136000153
000171000190000210000231000253000276000300^-1
I want the inverse of my sequence
number to look like this digit sequence.
So I then take this number to Wolfram Alpha (www.wolframalpha.com ) and find that it
is equal to
9.999970000029999990000000000000000000000000000000
000000...
× 10^11
I take the first 18 digits to form the
Sequence Number (because every digit after that is a zero):
999,997,000,002,999,999
Please notice that it looks very
similar to a sequence number used for Tribonacci sequences. The three parts are 999996, 1000002, and
999999 (the 6 in the first part and the 1 in the second part overlap and show
up as a 7 in the Sequence Number).
These suggest it is a 3, -3, 1 Tribonacci Sequence Number. This means we can produce a list of the
Triangular Numbers the way they are defined, but we can also produce them as
the 3, -3, 1 Tribonacci Sequence. Both
are the same.
But we still have not tested it to see
if it really works. It’s time to do
that now.
1/999997000002999999 = 1/999999^3 =
0.
000000 000000 000001 000003 000006 000010 000015 000021 000028 000036 000045 000055 000066 000078 000091 000105 000120 000136 000153 000171 000190 000210 000231 000253 000276 000300 000325 000351 000378 000406 000435 000465 000496 000528 000561 000595 000630 000666 000703 000741 000780 000820 000861 000903 000946 000990 001035 001081 001128 001176 001225 001275 001326 001378 001431 001485 001540 001596 001653 001711 001770 001830 001891 001953 002016 002080 002145 002211 002278 002346 002415 002485 002556 002628 002701 002775 002850 002926 003003 003081 003160 003240 003321 003403 003486 003570 003655 003741 003828 003916 004005 004095 004186 004278 004371 004465 004560 004656 004753 004851 004950 005050 005151 005253 005356 005460 005565 005671 005778 005886 005995 006105 006216 006328 006441 006555 006670 006786 006903 007021 007140 007260 007381 007503 007626 007750 007875 008001 008128 008256 008385 008515 008646 008778 008911 009045 009180 009316 009453 009591 009730 009870 010011 010153 010296 010440 010585 010731 010878 011026 011175 011325 011476 011628 011781 011935 012090 012246 012403 012561 012720 012880 013041 013203 013366 013530 013695 013861 014028 014196 014365 014535 014706 014878 015051 015225 015400 015576 015753 015931 016110 016290 016471 016653 016836 017020 017205 017391 017578 017766 017955 018145 018336 018528 018721 018915 019110 019306 019503 019701 019900 020100 020301 020503 020706 020910 021115 021321 021528 021736 021945 022155 022366 022578 022791 023005 023220 023436 023653 023871 024090 024310 024531 024753 024976 025200 025425 025651 025878 026106 026335 026565 026796 027028 027261 027495 027730 027966 028203 028441 028680 028920 029161 029403 029646 029890 030135 030381 030628 030876 031125 031375 031626 031878 032131 032385 032640 032896 033153 033411 033670 033930 034191 034453 034716 034980 035245 035511 035778 036046 036315 036585 036856 037128 037401 037675 037950 038226 038503 038781 039060 039340 039621 039903 040186 040470 040755 041041 041328 041616 041905 042195 042486 042778 043071 043365 043660 043956 044253 044551 044850 045150 045451 045753 046056 046360 046665 046971 047278 047586 047895 048205 048516 048828 049141 049455 049770 050086 050403 050721 051040 051360 051681 052003 052326 052650 052975 053301 053628 053956 054285 054615 054946 055278 055611 055945 056280 056616 056953 057291 057630 057970 058311 058653 058996 059340 059685 060031 060378 060726 061075 061425 061776 062128 062481 062835 063190 063546 063903 064261 064620 064980 065341 065703 066066 066430 066795 067161 067528 067896 068265 068635 069006 069378 069751 070125 070500 070876 071253 071631 072010 072390 072771 073153 073536 073920 074305 074691 075078 075466 075855 076245 076636 077028 077421 077815 078210 078606 079003 079401 079800 080200 080601 081003 081406 081810 082215 082621 083028 083436 083845 084255 084666 085078 085491 085905 086320 086736 087153 087571 087990 088410 088831 089253 089676 090100 090525 090951 091378 091806 092235 092665 093096 093528 093961 094395 094830 095266 095703 096141 096580 097020 097461 097903 098346 098790 099235 099681 100128 100576 101025 101475 101926 102378 102831 103285 103740 104196 104653 105111 105570 106030 106491 106953 107416 107880 108345 108811 109278 109746 110215 110685 111156 111628 112101 112575 113050 113526 114003 114481 114960 115440 115921 116403 116886 117370 117855 118341 118828 119316 119805 120295 120786 121278 121771 122265 122760 123256 123753 124251 124750 125250 125751 126253 126756 127260 127765 128271 128778 129286 129795 130305 130816 131328 131841 132355 132870 133386 133903 134421 134940 135460 135981 136503 137026 137550 138075 138601 139128 139656 140185 140715 141246 141778 142311 142845 143380 143916 144453 144991 145530 146070 146611 147153 147696 148240 148785 149331 149878 150426 150975 151525 152076 152628 153181 153735 154290 154846 155403 155961 156520 157080 157641 158203 158766 159330 159895 160461 161028 161596 162165 162735 163306 163878 164451 165025 165600 166176 166753 167331 167910 168490 169071 169653 170236 170820 171405 171991 172578 173166 173755 174345 174936 175528 176121 176715 177310 177906 178503 179101 179700 180300 180901 181503 182106 182710 183315 183921 184528 185136 185745 186355 186966 187578 188191 188805 189420 190036 190653 191271 191890 192510 193131 193753 194376 195000 195625 196251 196878 197506 198135 198765 199396 200028 200661 201295 201930 202566 203203 203841 204480 205120 205761 206403 207046 207690 208335 208981 209628 210276 210925 211575 ...
Written in six digit strings.
Accurate up to the 650th
non-zero term. 211,575 is the 650th
triangular number.
Compare with OEIS sequence A000217.
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Tetrahedral Numbers A000292
First terms
0, 1, 4, 10, 20, 35, 56, 84, 120, 165,
220, 286, 364, 455, 560, 680, 816,
969, 1140
Make 4 digit stings, take out comas and
spaces, add “0.” in front and
“^-1” behind.
0.00000001000400100020003500560084012001650220
02860364045505600680081609691140^-1 =
9.9960005999600010000000000000000000000000000000
00000
00000...
× 10^7
So it looks like the Sequence Number is 9,996,000,599,960,001. Let’s test it and
see if it works.
1/9996000599960001 =
0.
0000 0000 0000 0001 0004 0010 0020 0035 0056 0084 0120 0165 0220 0286 0364 0455 0560 0680 0816 0969 1140 1330 1540 1771 2024 2300 2600 2925 3276 3654 4060 4495 4960 5456 5984 6545 7140 7770 8436 9139 …
Written in four digit strings.
Accurate up to the 37th
non-zero term.
Compare with OEIS sequence A000292.
9996000599960001 looks like a
Tetranacci Sequence. It has four
parts: 9995, 10005, 9995, and 10001.
This translates into a 4,-6, 4, -1 Tetranacci Sequence: a(0) = a(1) =
a(2) = 0, a(3) = 1, and when n>3 then a(n) = 4*a(n-1) – 6*a(n-2) +
4*a(n-3) – 2*a(n-4). A quick spreadsheet
calculation confirms that the results are the same.
Can we do better?
9996000599960001 produces only four
digit terms.
9999996000000599999960000001 should
produce seven digit
terms.
1/9999996000000599999960000001 =
1/9999999^4 =
0.
0000000 0000000 0000000 0000001 0000004 0000010 0000020 0000035 0000056 0000084 0000120 0000165 0000220 0000286 0000364 0000455 0000560 0000680 0000816 0000969 0001140 0001330 0001540 0001771 0002024 0002300 0002600 0002925 0003276 0003654 0004060 0004495 0004960 0005456 0005984 0006545 0007140 0007770 0008436 0009139 0009880 0010660 0011480 0012341 0013244 0014190 0015180 0016215 0017296 0018424 0019600 0020825 0022100 0023426 0024804 0026235 0027720 0029260 0030856 0032509 0034220 0035990 0037820 0039711 0041664 0043680 0045760 0047905 0050116 0052394 0054740 0057155 0059640 0062196 0064824 0067525 0070300 0073150 0076076 0079079 0082160 0085320 0088560 0091881 0095284 0098770 0102340 0105995 0109736 0113564 0117480 0121485 0125580 0129766 0134044 0138415 0142880 0147440 0152096 0156849 0161700 0166650 0171700 0176851 0182104 0187460 0192920 0198485 0204156 0209934 0215820 0221815 0227920 0234136 0240464 0246905 0253460 0260130 0266916 0273819 0280840 0287980 0295240 0302621 0310124 0317750 0325500 0333375 0341376 0349504 0357760 0366145 0374660 0383306 0392084 0400995 0410040 0419220 0428536 0437989 0447580 0457310 0467180 0477191 0487344 0497640 0508080 0518665 0529396 0540274 0551300 0562475 0573800 0585276 0596904 0608685 0620620 0632710 0644956 0657359 0669920 0682640 0695520 0708561 0721764 0735130 0748660 0762355 0776216 0790244 0804440 0818805 0833340 0848046 0862924 0877975 0893200 0908600 0924176 0939929 0955860 0971970 0988260 1004731 1021384 1038220 1055240 1072445 1089836 1107414 1125180 1143135 1161280 1179616 1198144 1216865 1235780 1254890 1274196 1293699 1313400 1333300 1353400 1373701 1394204 1414910 1435820 1456935 1478256 1499784 1521520 1543465 1565620 1587986 1610564 1633355 1656360 1679580 1703016 1726669 1750540 1774630 1798940 1823471 1848224 1873200 1898400 1923825 1949476 1975354 2001460 2027795 2054360 2081156 2108184 2135445 2162940 2190670 2218636 2246839 2275280 2303960 2332880 2362041 2391444 2421090 2450980 2481115 2511496 2542124 2573000 2604125 2635500 2667126 2699004 2731135 2763520 2796160 2829056 2862209 2895620 2929290 2963220 2997411 3031864 3066580 3101560 3136805 3172316 3208094 3244140 3280455 3317040 3353896 3391024 3428425 3466100 3504050 3542276 3580779 3619560 3658620 3697960 3737581 3777484 3817670 3858140 3898895 3939936 3981264 4022880 4064785 4106980 4149466 4192244 4235315 4278680 4322340 4366296 4410549 4455100 4499950 4545100 4590551 4636304 4682360 4728720 4775385 4822356 4869634 4917220 4965115 5013320 5061836 5110664 5159805 5209260 5259030 5309116 5359519 5410240 5461280 5512640 5564321 5616324 5668650 5721300 5774275 5827576 5881204 5935160 5989445 6044060 6099006 6154284 6209895 6265840 6322120 6378736 6435689 6492980 6550610 6608580 6666891 6725544 6784540 6843880 6903565 6963596 7023974 7084700 7145775 7207200 7268976 7331104 7393585 7456420 7519610 7583156 7647059 7711320 7775940 7840920 7906261 7971964 8038030 8104460 8171255 8238416 8305944 8373840 8442105 8510740 8579746 8649124 8718875 8789000 8859500 8930376 9001629 9073260 9145270 9217660 9290431 9363584 9437120 9511040 9585345 9660036 9735114 9810580 9886435 …
Written in seven digit strings.
Accurate up to the 389th
non-zero term.
Compare with OEIS sequence A000292.
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The 3, -1, -1 Tribonacci Sequence
(written in six digit strings):
The 3, -1,-1 Tribonacci Sequence is
defined as: a(0) = a(1) = 0, a(2) = 1, and when n>2 then a(n) = 3*a(n-1) –
a(n-2) – a(n-3).
So let’s put together a Sequence Number
that we think will produce the 3, -1, -1 Tribonacci Sequence, written in six
digit strings.
The last of three groups should be
999999 + 1 (for subtracting 1*a(n-3)) + 1 more since it is the last group,
for a total of 1000001.
The middle group should be 999999 + 1
(for subtracting 1*a(n-2)) for a total of 1000000.
The first group should be 999999 – 3
(for adding 3*a(n-1)), for a total of 999996.
When you put the three parts together
(remembering to care to carry the seventh digits) 999996, 1000000, and
1000001 you get:
999,997,000,001,000,001
Let’s take it for a test drive and see
how it works.
1/999997000001000001 =
0.
000000 000000 000001 000003 000008 000020 000049 000119 000288 000696 001681 004059 009800 023660 057121 137903 332928 803761 940453
Since this sequence is not included in
the OIES database, we can run a quick spreadsheet simulation to check it:
It all looks good until the 16th
non-zero term. This is because the 17th
term has seven digits so it carries over and adds one to the 16th
term. And the 18th term
carries over to the 17th term … and so forth.
Written in six digit strings.
Accurate up to the 15th
non-zero term.
OEIS
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Back to the Beginning
The 2, -1 Fibonacci Sequence, written
in 3 digit strings, defined as: a(0) = 0, a(1) = 1, and when n>1 then a(n)
= 2*a(n-1) – a(n-2).
The last group should be 999 + 1 (for
subtracting a(n-2)) and + 1 again for being the last group. This gives us 1,001.
The first group should be 999 – 2 (for
adding 2*a(n-1)) which gives us 997.
Combining these two parts, being
careful to carry the extra digit from the last group to the first group gives
us:
998,001
But we have seen this Sequence Number
before, right at the very beginning of our romp through Sequence Numbers, and
we already know that it produces a counting sequence from “000” to “997” in
three digit strings.
Who knew that when kids learn to count
they are actually learning how to do a special case of the Fibonacci
Sequence!
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David
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