Friday, October 30, 2015

Shades of “Cantor’s Dust”

10/30/2015



Shades of “Cantor’s Dust”





The 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1 “27-nacci” Sequence (or just “27-nacci” Sequence):
Defined as: a(0) = … = a(25) = 0, a(26) = 1, and when n>26 then
a(n) = a(n-1) + a(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(11) + a(12) + a(n-13) + a(n-14)
+ a(n-15) + a(n-16) + a(n-17) + a(n-18) + a(n-19) + a(20) + a(n-21)
+ a(n-22) + a(n-23) + a(n-24) + a(n-25) + a(n-26) + a(n-27).
The Sequence Number is
999,999,899,999,989,999,998,999,999,899,999,989,999,
998,999,999,899,999,989,999,998,999,999,899,999,989,
999,998,999,999,899,999,989,999,998,999,999,899,999,
989,999,998,999,999,899,999,989,999,998,999,999,899,
999,989,999,998,999,999,899,999,989,999,999
1/999999899999989999998999999899999989999998999
99989999998999999899999989999998999999899999989
99999899999989999998999999899999989999998999999
89999998999999899999989999998999999899999989999
999 =
0.
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000001  0000001  0000002  0000004 
0000008  0000016  0000032  0000064  0000128  0000256 
0000512  0001024  0002048  0004096  0008192  0016384 
0032768  0065536  0131072  0262144  0524288  1048576 
2097152  4194304 
Terms are written in seven digit strings.
Terms are accurate up to the 24th non-zero term.
This sequence is not listed in the OEIS – yet!
The 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1,
1, 1, 1, 1, 1 “27-nacci” Sequence:
Defined as: a(0) = … = a(25) = 0, a(26) = 1, and when n>26
then a(n) = a(n-1) + a(2) + a(n-3) + a(n-4) + a(n-5) + a(n-6)
+ a(n-7) + a(n-8) + a(n-9) + a(n-19) + a(20) + a(n-21) + a(n-22)
+ a(n-23) + a(n-24) + a(n-25) + a(n-26) + a(n-27).
The Sequence Number is
999,999,899,999,989,999,998,999,999,899,999,989,999,
998,999,999,899,999,989,999,998,999,999,999,999,999,
999,999,999,999,999,999,999,999,999,999,999,999,999,
999,999,999,999,999,899,999,989,999,998,999,999,899,
999,989,999,998,999,999,899,999,989,999,999
1/999999899999989999998999999899999989999998999
99989999998999999899999999999999999999999999999
99999999999999999999999999999999999999998999999
89999998999999899999989999998999999899999989999
999 =
0.
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000001  0000001  0000002  0000004 
0000008  0000016  0000032  0000064  0000128  0000256 
0000511  0001021  0002040  0004076  0008144  0016272 
0032512  0064960  0129792  0259329  0518148  1035277 
2068518  4132968 
Terms are written in seven digit strings.
Terms are accurate up to the 24th non-zero term.
This sequence is not listed in the OEIS collection.
The 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 1, 1, 1 “27-nacci” Sequence:
Defined as: a(0) = … = a(25) = 0, a(26) = 1, and when n>26
then a(n) = a(n-1) + a(2) + a(n-3) + a(n-7) + a(n-8) + a(n-9)
+ a(n-19) + a(20) + a(n-21) + a(n-25) + a(n-26) + a(n-27).
The Sequence Number is
999,999,899,999,989,999,998,999,999,999,999,999,999,
999,999,999,899,999,989,999,998,999,999,999,999,999,
999,999,999,999,999,999,999,999,999,999,999,999,999,
999,999,999,999,999,899,999,989,999,998,999,999,999,
999,999,999,999,999,999,899,999,989,999,999
1/999999899999989999998999999999999999999999999
99989999998999999899999999999999999999999999999
99999999999999999999999999999999999999998999999
89999998999999999999999999999999999899999989999
999 =
0.
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000001  0000001  0000002  0000004  0000007  0000013  0000024  0000045  0000084  0000157 
0000293  0000547  0001021  0001905  0003555  0006634  0012380  0023103  0043114  0080459  0150151  0280209 
0522920  0975862  1821132  3398556 
Terms are written in seven digit terms.
Terms are accurate up to the 26th non-zero term.
This sequence is not listed in the OEIS collection.
The 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 1, 0, 1 “27-nacci” Sequence:
Defined as: a(0) = … = a(25) = 0, a(26) = 1, and when n>26
then a(n) = a(n-1) + a(n-3) + a(n-7) + a(n-9) + a(n-19) +
a(n-21) + a(n-25) + a(n-27).
The Sequence Number is
999,999,899,999,999,999,998,999,999,999,999,999,999,
999,999,999,899,999,999,999,998,999,999,999,999,999,
999,999,999,999,999,999,999,999,999,999,999,999,999,
999,999,999,999,999,899,999,999,999,998,999,999,999,
999,999,999,999,999,999,899,999,999,999,999
1/999999899999999999998999999999999999999999999
99989999999999999899999999999999999999999999999
99999999999999999999999999999999999999998999999
99999998999999999999999999999999999899999999999
999 =
0.
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000 
0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000000  0000001  0000001  0000001  0000002  0000003  0000004  0000006  0000010  0000015  0000023 
0000036  0000055  0000084  0000129  0000198  0000303  0000465  0000714  0001095  0001681  0002580  0003959 
0006075  0009322  0014304  0021949  0033681  0051683  0079308  0121699  0186747  0286563  0439731  0674766 
1035426  1588860  2438103  3741264  5740965 
Terms are written in seven digit strings.
Terms are accurate up to the 39th non-zero term.
This sequence is not listed in the OEIS collection – yet!
Since OEIS does not list any of the sequences shown above I developed
a spreadsheet to calculate the terms in each of the sequences showing all
terms with seven digits or less.  You can compare the sequences shown
above (calculated via Sequence Number) to the terms shown below
(calculated by recursive calculations in a spreadsheet).  You will not
hurt my feelings if you decide to do the calculations yourself – I did
the same thing when I first saw a Sequence Number on the web.  I
was skeptical and did the calculations myself.
The 27-nacci
Sequence
The 1111111110000-
00000111111111
27-nacci
Sequence
The
1110001110000-
00000111000111
27-nacci
Sequence
The 1010001010000-
00000101000101
27-nacci
Sequence
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
2
2
2
1
4
4
4
2
8
8
7
3
16
16
13
4
32
32
24
6
64
64
45
10
128
128
84
15
256
256
157
23
512
511
293
36
1,024
1,021
547
55
2,048
2,040
1,021
84
4,096
4,076
1,905
129
8,192
8,144
3,555
198
16,384
16,272
6,634
303
32,768
32,512
12,380
465
65,536
64,960
23,103
714
131,072
129,792
43,114
1,095
262,144
259,329
80,459
1,681
524,288
518,148
150,151
2,580
1,048,576
1,035,277
280,209
3,959
2,097,152
2,068,518
522,920
6,075
4,194,304
4,132,968
975,862
9,322
8,388,608
8,257,808
1,821,132
14,304
16,777,216
16,499,376
3,398,556
21,949
33,554,432
32,966,304
6,342,310
33,681
67,108,864
65,867,776
11,835,879
51,683


22,087,857
79,308


41,219,873
121,699


76,923,621
186,747



286,563



439,731



674,766



1,035,426



1,588,860



2,438,103



3,741,264



5,740,965



8,809,503



13,518,168



20,743,608



31,831,035



48,844,671



74,952,069

These Sequence Numbers are the largest (longest) that I have been able to calculate with so far.  I was able to get seven digit strings, but when I tried to get eight digit strings the Sequence Number was to long for me to use on Wolfram Alpha.  I'm not complaining, not many calculations need to use an input of 200 or more digits, or an output of 3900 digits..

Did you catch the comparison of these sequences with “Cantor’s Dust”?


David


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