A005203 Fibonacci numbers (or rabbit sequence) converted to decimal. (Formerly M1539)

0, 1, 2, 5, 22, 181, 5814, 1488565, 12194330294, 25573364166211253, 439347050970302571643057846, 15829145720289447797800874537321282579904181, 9797766637414564027586288536574448245991597197836000123235901011048118

Offset

0,3

Comments

a(n) is also the denominator of the continued fraction [2^F(0), 2^F(1), 2^F(2), 2^F(3), 2^F(4), ..., 2^F(n-1)] for n>0. For the numerator, see A063896. - Chinmay Dandekar and Greg Dresden, Sep 11 2020

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..17

J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc., 63 (1977), 29-32.

H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., Sequences associated with t-ary coding of Fibonacci's rabbits, Fib. Quart., 15 (1977), 311-318.

Ron Knott, The Fibonacci Rabbit Sequence

Ron Knott, Rabbit Sequence in Zeckendorf Expansion (A003714)

Eric Weisstein's World of Mathematics, Rabbit Sequence

Formula

a(0) = 0, a(1) = 1, a(n) = a(n-1) * 2^F(n-1) + a(n-2).

a(n) = rewrite_0to1_1to10_n_i_times(0, n) [ Each 0->1, 1->10 in binary expansion ]

Maple

rewrite_0to1_1to10_n_i_times := proc(n, i) local z, j; z := n; j := i; while(j > 0) do z := rewrite_0to1_1to10(z); j := j - 1; od; RETURN(z); end;
rewrite_0to1_1to10 := proc(n) option remember; if(n < 2) then RETURN(n + 1); else RETURN(((2^(1+(n mod 2))) * rewrite_0to1_1to10(floor(n/2))) + (n mod 2) + 1); fi; end;

Mathematica

a[0] = 0; a[1] = 1; a[n_] := a[n] = a[n-1]*2^Fibonacci[n-1] + a[n-2]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jul 27 2011 *)

Crossrefs

Cf. A000045, A048707, A003714, A048721, A048722, A048678, A048679, A048680, A005205, A063896.

Column k=2 of A144287.

Sequence in context: A361331 A342967 A042933 * A193660 A090450 A137099

Adjacent sequences: A005200 A005201 A005202 * A005204 A005205 A005206

Keyword

nonn,base

Author

N. J. A. Sloane

Extensions

Comments and more terms from Antti Karttunen, Mar 30 1999

Status

approved