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A005203

Fibonacci numbers (or rabbit sequence) converted to decimal.

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

(list; graph; refs; listen; history; text; internal format)

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 *)

PROG

(Python)

def A005203(n):

s = '0'

for i in range(n):

s = s.replace('0', 'a').replace('1', '10').replace('a', '1')

return int(s, 2) # Chai Wah Wu, Apr 24 2025

CROSSREFS

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

Column k=2 of A144287.