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