A005205 -id:A005205 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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

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

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

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

Column k=2 of A144287.

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;

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 *)

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

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 ]

Comments and more terms from Antti Karttunen, Mar 30 1999

A144287 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) = Fibonacci rabbit sequence number n coded in base k.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 10, 22, 5, 1, 5, 17, 93, 181, 8, 1, 6, 26, 276, 2521, 5814, 13, 1, 7, 37, 655, 17681, 612696, 1488565, 21, 1, 8, 50, 1338, 81901, 18105620, 4019900977, 12194330294, 34, 1, 9, 65, 2457, 289045, 255941280, 1186569930001, 6409020585966267, 25573364166211253, 55

Offset: 1

Alois P. Heinz, Sep 17 2008

			Square array begins:
  1,   1,    1,     1,     1,  ...
  1,   2,    3,     4,     5,  ...
  2,   5,   10,    17,    26,  ...
  3,  22,   93,   276,   655,  ...
  5, 181, 2521, 17681, 81901,  ...

Alois P. Heinz, Antidiagonals n = 1..16, flattened
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.

Columns k=1-10 give: A000045, A005203, A005205, A320986, A320987, A320988, A320989, A320990, A320991, A061107 and A036299.
Rows n=1-3 give: A000012, A001477, A002522.
Main diagonal gives A144288.

f:= proc(n,b) option remember; `if`(n<2, [n,n], [f(n-1, b)[1]*
       b^f(n-1, b)[2] +f(n-2, b)[1], f(n-1, b)[2] +f(n-2, b)[2]])
    end:
A:= (n,k)-> f(n,k)[1]:
seq(seq(A(n, 1+d-n), n=1..d), d=1..11);

f[n_, b_] := f[n, b] = If[n < 2, {n, n}, {f[n-1, b][[1]]*b^f[n-1, b][[2]] + f[n-2, b][[1]], f[n-1, b][[2]] + f[n-2, b][[2]]}]; t[n_, k_] := f[n, k][[1]]; Flatten[ Table[t[n, 1+d-n], {d, 1, 11}, {n, 1, d}]] (* Jean-François Alcover, translated from Maple, Dec 09 2011 *)

See program.

A144288 Fibonacci rabbit sequence number n coded in base n, also diagonal of A144287.

Original entry on oeis.org

1, 2, 10, 276, 81901, 2247615258, 81658169024988865, 644986443956439734064225751112, 3427833941153173630835645403655873661712817810325122

Offset: 1

Alois P. Heinz, Sep 17 2008

Alois P. Heinz, Table of n, a(n) for n = 1..14

Cf. A000045, A005203, A005205, A061107, A036299, A144287.

f:= proc(n, b) option remember; `if`(n<2, [n, n], [f(n-1, b)[1] *b^f(n-1, b)[2] +f(n-2, b)[1], f(n-1, b)[2] +f(n-2, b)[2]]) end: a:= n-> f(n, n)[1]: seq(a(n), n=1..11);

f[n_, b_] := f[n, b] = If[n < 2, {n, n}, {f[n-1, b][[1]]*b^f[n-1, b][[2]] + f[n-2, b][[1]], f[n-1, b][[2]] + f[n-2, b][[2]]}]; a[n_] := f[n, n][[1]]; Table[a[n], {n, 1, 9}] (* Jean-François Alcover, Jan 03 2013, translated from Maple *)

See program.

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

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A144287 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) = Fibonacci rabbit sequence number n coded in base k.

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A144288 Fibonacci rabbit sequence number n coded in base n, also diagonal of A144287.

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