A048721 -id:A048721 - 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

A048722 Reversed binary packing of Fibonacci sequence A000045.

Original entry on oeis.org

0, 1, 3, 7, 29, 233, 7457, 1908993, 15638470657, 32796250015268865, 563435284988077103288156161, 20299895516157546089301785397257605216206849, 12565026726380593749379544715414757684521993402717913413208480665305089

Offset: 0

Antti Karttunen, Mar 30 1999

nonn

Cf. A000045, A005203, A048721, A036571.

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

A036571 Binary packing of Connell sequence (shifted once right).

Original entry on oeis.org

0, 1, 3, 11, 27, 91, 347, 859, 2907, 11099, 43867, 109403, 371547, 1420123, 5614427, 22391643, 55946075, 190163803, 727034715, 2874518363, 11464452955, 45824191323, 114543668059, 389421575003

Offset: 0

Antti Karttunen

nonn

Binary representation of n has 1's at positions specified by Connell sequence (A001614).

			347=101011011 in binary, with 1's at positions 1,2,4,5,7,9.

Cf. A001614, A048721, A048722.

from itertools import count, islice
from math import isqrt
def A036571_gen(): # generator of terms
    c = 0
    for n in count(1):
        yield c
        c += 1<<(m:=n<<1)-(k:=isqrt(m))-int((m<<2)>(k<<2)*(k+1)+1)-1
A036571_list = list(islice(A036571_gen(),25)) # Chai Wah Wu, Jul 26 2022

a(0)=0, a(n) = a(n-1) + 2^((2*n - floor((1/2)*(1 + sqrt(8*n - 7)))) - 1).

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

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A048722 Reversed binary packing of Fibonacci sequence A000045.

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A036571 Binary packing of Connell sequence (shifted once right).

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