A111061 Begin with 1,2 In binary 1, 10. To get the sequence, left pad binary number with its precedent: 1,10, 110, 10110, 11010110, 1011011010110, etc. Note the number of bits of the n-th term is the (n-1)st Fibonacci number. Now convert back to decimal 1,2,6,22,214,5846, ...

1, 2, 6, 22, 214, 5846, 1758934, 12261709526, 30218268284999382, 441774643647969157361358550, 18704202113934148330876898021651431451973334, 9851903763165025237741730894918087846312835864942483209357642906130134

Offset: 1

Author

Rosario Trifiletti (rtrif(AT)aol.com), Oct 07 2005

Keywords

• nonn
• base

Comments

Another way to represent these numbers is as a binary pyramid

1

10

110

10110

11010110

1011011010110

110101101011011010110

1011011010110110101101011011010110

110101101011011010110101101101011011010110101101101011010110110101101101011010110110101101101011010110110101101011011010110110101101011011010110

Obviously 10110 plays a key role in this sequence ... Call M(n) the n-term in the sequence M(n)clearly increases monotonically but M(n+1)/M(n) does not. The first few values of M(n+1)/M(n) are : 2 3 35.666.. 27.31775701 300.8782073 6971.102683 2464441.701 512 Does this converge? What to? I propose calling them FIFO-nacci numbers ...

The length of each binary term above is a Fibonacci number (A000045). The number of decimal digits is: 1, 1, 1, 2, 3, 4, 7, 11, 17, 27, 44, 70, 114, 184, 298, 481, 778, 1259, 2037, 3295, 5332, 8627, 13959, 22585, 36543, 59128, 95671, 154799, 250469, 405268, 655737, 1061004, 1716740, 2777744, 4494484, 7272228, 11766712, ..., . - Robert G. Wilson v, Aug 24 2007

Programs

Mathematica

f[l_] := Append[l, FromDigits[ Join[ IntegerDigits[l[[ -2]], 2], IntegerDigits[l[[ -1]], 2]], 2]]; Nest[f, {1, 2}, 10] (* Robert G. Wilson v, Aug 24 2007 *)

Extensions

More terms from Robert G. Wilson v, Aug 24 2007