The Octanacci and Super-Octanacci Sequences

 7/04/2015

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). The decimal expansion of the inverse of 999,999,998,999,999,998,999, 999,998,999,999,998,999,999, 998,999,999,998,999,999,998, 999,999,999 will produce a sequence that shows the terms of the Octanacci Numbers. 1/99999999899999999899999 9998999999998999999998999 999998999999998999999999 = 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) = 1and 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). The decimal expansion of the inverse of 999,999,998,999,999,997,999, 999,996,999,999,995,999,999, 994,999,999,993,999,999,992, 999,999,992 will produce a sequence that shows the terms of the Super-Octanacci Numbers. 1/99999999899999999799999 9996999999995999999994999 999993999999992999999992 = 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 …

David