## Tuesday, June 2, 2015

### The Fibonacci and Super-Fibonacci Sequences

6/02/2015

The Fibonacci Sequence: a(0) = 0, a(1) = 1, an when n>1 then a(n) = a(n-1) + a(n-2).  OEIS A000045.
The digital expansion of the inverse of 999,999,998,999,999,999 produces the first 43 terms of the Fibonacci Sequence in nine digit strings.
1/999999998999999999 =
0.
000000000 000000001 000000001 000000002 000000003 000000005 000000008 000000013 000000021 000000034 000000055 000000089 000000144 000000233 000000377 000000610 000000987 000001597 000002584 000004181 000006765 000010946 000017711 000028657 000046368 000075025 000121393 000196418 000317811 000514229 000832040 001346269 002178309 003524578 005702887 009227465 014930352 024157817 039088169 063245986 102334155 165580141 267914296 433494437 …

The Super Fibonacci Sequence: a(0) = 0, a(1) = 1, an when n>1 then a(n) = a(n-1) + 2*a(n-2).
The digital expansion of the inverse of 999,999,998,999,999,998 produces the first 30 terms of the Fibonacci Sequence in nine digit strings.
1/999999998999999998 =
0.
000000000 000000001 000000001 000000003 000000005 000000011 000000021 000000043 000000085 000000171 000000341 000000683 000001365 000002731 000005461 000010923 000021845 000043691 000087381 000174763 000349525 000699051 001398101 002796203 005592405 011184811 022369621 044739243 089478485 178956971 357913941 …

David

1. bonus fact: 999999998999999999 is a prime!

2. Super Fibonacci Sequence is Jacobsthal Sequence