Undecanacci and Super-Undecanacci Sequences

6/11/2015

The Undecanacci 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).

The decimal expansion of the inverse of 999,999,998,999,999,998,999,999,998,999,999,998,999,999,
998,999,999,998,999,999,998,999,999,998,999,999,998,999,
999,998,999,999,999 produces a sequence of undecanacci numbers written in 9 digit strings.

1/999999998999999998999999998999999998999999998
99999999899999999899999999899999999899999999899
9999999 =

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) = 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).

The  decimal expansion of  the inverse of 999,999,998,999,999,997,999,999,996,999,999,995,999,999,
994,999,999,993,999,999,992,999,999,991,999,999,990,999,
999,989,999,999,989 produces a sequence of super-undecanacci number written in 9 digit strings.

1/999999998999999997999999996999999995999999994
99999999399999999299999999199999999099999998999
9999989 =

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 …



How do they know how to do that???



David