A063896 -id:A063896 - 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

A061107 a(0) = 0, a(1) = 1, a(n) is the concatenation of a(n-2) and a(n-1) for n > 1.

Original entry on oeis.org

0, 1, 10, 101, 10110, 10110101, 1011010110110, 101101011011010110101, 1011010110110101101011011010110110, 1011010110110101101011011010110110101101011011010110101

Offset: 0

Amarnath Murthy, Apr 20 2001

Original name was: In the Fibonacci rabbit problem we start with an immature pair 'I' which matures after one season to 'M'. This mature pair after one season stays alive and breeds a new immature pair and we get the following sequence I, MI, MIM, MIMMI, MIMMIMIM, MIMMIMIMMIMMI... if we replace 'I' by a '0' and 'M' by a '1' we get the required binary sequence.

			a(0) = 0, a(1) = 1, a(2) = a(1)a(0)= 10, etc.

Amarnath Murthy, Smarandache Reverse auto correlated sequences and some Fibonacci derived sequences, Smarandache Notions Journal Vol. 12, No. 1-2-3, Spring 2001.
Ian Stewart, The Magical Maze.

Harry J. Smith, Table of n, a(n) for n = 0..15
Duaa Abdullah and Jasem Hamoud, Dynamic and Programmatic Analysis of Fibonacci Word Density, arXiv:2504.04087 [math.CO], 2025. See p. 10.

Cf. A063896, A131242. See A005203 for the sequence version converted to decimal.

Column k=10 of A144287.

A[0]:= 0: A[1]:= 1: A[2]:= 10:
for n from 3 to 20 do
A[n]:= 10^(ilog10(A[n-2])+1)*A[n-1]+A[n-2]
od:
seq(A[n],n=0..10); # Robert Israel, Apr 30 2015

nxt[{a_,b_}]:={b,FromDigits[Join[IntegerDigits[b],IntegerDigits[a]]]}; Transpose[NestList[nxt,{0,1},10]][[1]] (* Harvey P. Dale, Jul 05 2015 *)

{ default(realprecision, 100); L=log(10); for (n=0, 15, if (n>2, a=a1*10^(log(a2)\L + 1) + a2; a2=a1; a1=a, if (n==0, a=0, if (n==1, a=a2=1, a=a1=10))); write("b061107.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 18 2009

a(0) = 0, a(1) =1, a(n) = concatenation of a(n-1) and a(n-2).
a(n) = a(n-1)*2^floor(log_2(a(n-2))+1)+a(n-2), for n>2, a(2)=10 (base 2). - Hieronymus Fischer, Jun 26 2007
a(n) = A036299(n-1), n>0. - R. J. Mathar, Oct 02 2008
a(n) can be transformed by a(n-1) when you change every single "1"(from a(n-1)) into "10" and every single "0"(from a(n-1)) into "1". [YuJiping and Sirius Caffrey, Apr 30 2015]

More terms from Hieronymus Fischer, Jun 26 2007

A131293 Concatenate a(n-2) and a(n-1) to get a(n); start with a(0)=0, a(1)=1, delete the leading zero. Also: concatenate Fibonacci(n) 1's.

Original entry on oeis.org

0, 1, 1, 11, 111, 11111, 11111111, 1111111111111, 111111111111111111111, 1111111111111111111111111111111111, 1111111111111111111111111111111111111111111111111111111

Offset: 0

Hieronymus Fischer, Jun 26 2007

Interpreted as base-2 numbers the result is A063896.

This sequence differs from A108047 by the leading a(0) = 0. - Jason Kimberley, Dec 15 2012

			a(3)=11, a(4)=111, so a(5) = a(4)*a(3) = 11111.

Vincenzo Librandi, Table of n, a(n) for n = 0..18

Cf. A000045, A061107.

Cf. A037842, A108047.

import Data.Function (on)
a131293 n = a131293_list !! n
a131293_list = 0 : 1 : map read
               (zipWith ((++) `on` show) a131293_list $ tail a131293_list)
-- Reinhard Zumkeller, Oct 05 2015

[(10^Fibonacci(n)-1)/9: n in [0..10]]; // Vincenzo Librandi, Aug 29 2011

a:= n-> parse(cat(0, 1$combinat[fibonacci](n))):
seq(a(n), n=0..11);  # Alois P. Heinz, Apr 17 2020

Join[{0},FromDigits/@(PadLeft[{},#,1]&/@Fibonacci[Range[10]])] (* Harvey P. Dale, Aug 28 2011 *)

a(n) = a(n-2)*10^ceiling(log_10(a(n-1))) + a(n-1) for n > 1.

a(n) = (10^Fibonacci(n) - 1)/9.

A254132 a(0)=1 and a(1)=2, then each term is x + y + x*y where x and y are the 2 last terms.

Original entry on oeis.org

1, 2, 5, 17, 107, 1943, 209951, 408146687, 85691213438975, 34974584955819144511487, 2997014624388697307377363936018956287, 104819342594514896999066634490728502944926883876041385836543

Offset: 0

Michel Marcus, Jan 26 2015

nonn

			a(0) = 1, a(1) = 2, a(2) = 1+2+(1*2) = 5, a(3) = 2+5+(2*5) = 17.

Ana Rechtman, Janvier 2015, 3ème défi, (in French), Images des Mathématiques, CNRS, 2015.
Ana Rechtman, Solution, (in French), Images des Mathématiques, CNRS, 2015.

Cf. A000045 (Fibonacci), A063896 (similar, with initial values 0,1).

Cf. A198796 (2^n*3^(n+1)-1).

a254132[0]=1;a254132[n_]:=2^Fibonacci[n-1]*3^Fibonacci[n]-1;
a254132/@Range[0,11] (* Ivan N. Ianakiev, Jan 27 2015 *)

lista(nn) = {x = 1; y = 2; print1(x, ", ", y, ", "); for (j=1, nn, z = x + y + x*y; print1(z, ", "); x = y; y = z;);}

a(n) = if (!n, 1, 2^fibonacci(n)*3^fibonacci(n+1) - 1);

a(n) = a(n-1) + a(n-2) + a(n-1)*a(n-2).
a(0) = 1 and a(n) = 2^Fibonacci(n)*3^Fibonacci(n+1) - 1 (see 2nd link).
a(n) == 8 mod 9, for n > 2. - Ivan N. Ianakiev, Jan 27 2015

A073128 Integer quotients arising in A063986.

Original entry on oeis.org

0, 1, 1, 5, 5, 49, 118, 121, 2406, 2698, 4182, 4946, 153627, 3087192, 8203485, 38394376, 487844934, 2822741576, 4140154385, 4397137572, 8583966231

Offset: 1

Labos Elemer, Jul 16 2002

			n=15, A063986(15)=41846733, sum of first 41846733 cototients is s=343289046464505, a(15)=s/41846733. Only in knowledge of either the quotients or of partial sums of cototients is it possible to continue A063986 without recomputing previous terms!

Cf. A051953, A000010, A002088, A048920, A000217, A050226, A063986.

a(n)=Sum[cototient[j], j=1..A063986(n)]/A063986(n)

Changed A063896 to A063986 and a(16)-a(21) from Donovan Johnson, May 11 2010

A073129 Partial sums of cototients arising in A063986.

Original entry on oeis.org

0, 4, 5, 120, 125, 12201, 70800, 74657, 29526432, 37132574, 89210424, 124777688, 120392102955, 48617257062792, 343289046464505, 7519663359716376, 1214022599940709056, 40644839476190305216, 87437200646372849005, 98628693371623948080, 375871306587181970568

Offset: 1

Labos Elemer, Jul 16 2002

nonn

			n=15, a(15)=343289046464505, sum of first 41846733 cototients A063986(15)*A073128(15)=a(15) To continue A063986, A073128 or A073129 without recomputing previous terms, corresponding entries from 2 of above sequences is required.

Cf. A051953, A000010, A002088, A048920, A000217, A050226, A073128, A063986.

a(n)=Sum[cototient[j], j=1..A063986(n)]

Changed A063896 to A063986 and a(16)-a(21) from Donovan Johnson, May 11 2010

A100701 a(n) = a(n-1) + a(n-2) + a(n-1)*a(n-2) for n>=2; a(0)=2, a(1)=3.

Original entry on oeis.org

2, 3, 11, 47, 575, 27647, 15925247, 440301256703, 7011906707722862591, 3087351335301583621409910816767, 21648219537098310851336266290644502090473753542655

Offset: 0

Parthasarathy Nambi, Dec 09 2004

nonn

In general, if b(n) is defined recursively by b(0) = p, b(1) = q, b(n) = b(n-1) + b(n-2) + b(n-1) * b(n-2) for n >= 2 then b(n) = p^Fibonacci(n-1) * q^Fibonacci(n) - 1. - Rahul Goswami, Apr 15 2020

			a(2) = (2 + 3) + 2*3 = 11.

Cf. A000045 (Fibonacci), A063896.

a=2;b=3;lst={a,b};Do[c=a*b+a+b;AppendTo[lst,c];a=b;b=c,{n,2*3!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009 *)

a(n)={3^fibonacci(n-1) * 4^fibonacci(n) - 1} \\ Andrew Howroyd, Apr 14 2020

a(n) = a(n-1) + a(n-2) + a(n-1)*a(n-2) with a(0)=2 and a(1)=3.

a(n) = 3^Fibonacci(n-1) * 4^Fibonacci(n) - 1. - Rahul Goswami, Apr 15 2020

More terms from Vladimir Joseph Stephan Orlovsky, Sep 05 2009

A121389 a(n) = 10^Fibonacci(n) - 1.

Original entry on oeis.org

0, 9, 9, 99, 999, 99999, 99999999, 9999999999999, 999999999999999999999, 9999999999999999999999999999999999, 9999999999999999999999999999999999999999999999999999999

Offset: 0

Rick L. Shepherd, Jul 26 2006

nonn

Each a(n) has Fibonacci(n) (trailing) 9's. In general, if the same recurrence below is used with any a(0), a(1) >= 0, then, for all k >= 2, a(k) has the same number of trailing 9's as a(k-2) and a(k-1) have altogether (see for example A121390).

Cf. A000045, A063896, A108047, A121390.

10^Fibonacci[Range[0,10]]-1 (* Harvey P. Dale, Dec 24 2022 *)

a(n) = 10^Fibonacci(n) - 1 = 10^A000045(n) - 1 (= 9*A108047(n) for n>=1). a(0) = 0; a(1) = 9; a(n) = a(n-2)*a(n-1) + a(n-2) + a(n-1).

A121390 Sequence generated by the recurrence below begins with exactly seven primes.

Original entry on oeis.org

2, 8607769, 25823309, 222281113118699, 5740034091209256896999, 1275901167133279185594625154673899999, 7323716196358702385046852882006552347670105360888299999999

Offset: 0

Rick L. Shepherd, Jul 26 2006

nonn

This sequence has the smallest value of a(1) for a(0) = 2 and this recurrence such that the first seven terms are primes. The first composite term is too large to include. Note also that each a(n) has Fibonacci(n) trailing 9's (see A121389).

			a(2) = 25823309 = 2*8607769 + 2 + 8607769 and is prime.

Cf. A000045, A063896, A121389.

a(0) = 2; a(1) = 8607769; a(n) = a(n-2)*a(n-1) + a(n-2) + a(n-1).

A333453 Binary concatenation (ignoring leading zeros) of a(n-1) and a(n-2) mod n, starting with a(n) = n for n <= 1.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 3, 0, 3, 3, 5, 1, 1, 3, 7, 1, 15, 14, 5, 18, 9, 12, 3, 14, 11, 15, 17, 17, 1, 20, 11, 0, 11, 11, 17, 3, 5, 23, 37, 37, 5, 29, 27, 33, 27, 6, 35, 4, 3, 28, 15, 49, 19, 46, 33, 13, 25, 14, 9, 40, 49, 4, 57, 19, 57, 23, 11, 40, 39, 52, 7, 3, 31

Offset: 0

Alois P. Heinz, Mar 21 2020

Value 0 is treated as empty bit string.

			a(18) = 5 = 239 mod 18, where 239 = 11101111_2 is the binary concatenation 1110_2 = 14 = a(17) and 1111_2 = 15 = a(16).

Alois P. Heinz, Table of n, a(n) for n = 0..32768

Cf. A062383, A063896, A079777.

a:= proc(n) option remember; `if`(n<2, n, (t-> a(n-1)*
      `if`(t=0, 1, 2^(ilog2(t)+1))+t)(a(n-2)) mod n)
    end:
seq(a(n), n=0..100);

a(n) = (a(n-1)*A062383(a(n-2)) + a(n-2)) mod n if n > 1, a(n) = n if n < 2.

cp's OEIS Frontend

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

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A061107 a(0) = 0, a(1) = 1, a(n) is the concatenation of a(n-2) and a(n-1) for n > 1.

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A131293 Concatenate a(n-2) and a(n-1) to get a(n); start with a(0)=0, a(1)=1, delete the leading zero. Also: concatenate Fibonacci(n) 1's.

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A254132 a(0)=1 and a(1)=2, then each term is x + y + x*y where x and y are the 2 last terms.

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Mathematica

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PARI

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A073128 Integer quotients arising in A063986.

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A073129 Partial sums of cototients arising in A063986.

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A100701 a(n) = a(n-1) + a(n-2) + a(n-1)*a(n-2) for n>=2; a(0)=2, a(1)=3.

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