A144287 -id:A144287 - cp's OEIS frontend

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

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.

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

Original entry on oeis.org

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

A036299 Binary Fibonacci (or rabbit) sequence.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

A055642(a(n)) = A000045(n+2). - Reinhard Zumkeller, Jul 06 2014

N. G. De Bruijn, (1989, January). Updown generation of Beatty sequences. Koninklijke Nederlandsche Akademie van Wetenschappen (Indationes Math.), Proc., Ser. A, 92:4 (1968), 385-407. See Fig. 3.
J. Kappraff, D. Blackmore and G. Adamson, Phyllotaxis as a dynamical system: a study in number, Chap. 17 of Jean and Barabe, eds., Symmetry in Plants, World Scientific, Studies in Math. Biology and Medicine, Vol. 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..14
M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers Math. Applic., 29 (No, 12, 1995), 103-110.
C. J. Glasby, S. P. Glasby and F. Pleijel, Worms by number, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
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.

Cf. A005614, A003849.

Column k=10 of A144287.

a036299 n = a036299_list !! n
a036299_list = map read rabbits :: [Integer] where
   rabbits = "1" : "10" : zipWith (++) (tail rabbits) rabbits
-- Reinhard Zumkeller, Jul 06 2014

nxt[{a_,b_}]:=FromDigits[Join[IntegerDigits[b],IntegerDigits[a]]]; Transpose[NestList[{Last[#],nxt[#]}&,{1,10},10]][[1]] (* Harvey P. Dale, Oct 16 2011 *)

def aupton(terms):
  alst = [1, 10]
  while len(alst) < terms: alst.append(int(str(alst[-1]) + str(alst[-2])))
  return alst[:terms]
print(aupton(9)) # Michael S. Branicky, Jan 10 2021

a(n+1) = concatenation of a(n) and a(n-1).

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

A005205 Coding Fibonacci numbers.

Original entry on oeis.org

1, 3, 10, 93, 2521, 612696, 4019900977, 6409020585966267, 67040619014505181883304178, 1118048584563024433220786501983631190591549, 195042693446883195450571898296824337898272003171567594807867055549521

Offset: 1

N. J. A. Sloane

Binary Fibonacci (or rabbit) sequence A036299, read in base 3, then converted to decimal. - Jonathan Vos Post, Oct 19 2007

			a(0) = 1 because A036299(0) = "1" and 1 base 3 = 1 base 10.
a(1) = 3 because A036299(1) = "10" and 10 base 3 = 3 base 10.
a(2) = 10 because A036299(2) = "101" and 101 base 3 = 10 base 10.
a(3) = 93 because A036299(3) = "10110" and 10110 base 3 = 93 base 10.
a(4) = 2521 because A036299(4) = "10110101" and 10110101 base 3 = 2521 base 10.
a(5) = 612696 because A036299(5) = "1011010110110" and 1011010110110 base 3 = 612696 base 10.

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

Alois P. Heinz, Table of n, a(n) for n = 1..16
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.

Cf. A005203, A036299.

Column k=3 of A144287.

b:= proc(n) option remember; `if` (n<2, [n, n], [b(n-1)[1] *3^b(n-1)[2] +b(n-2)[1], b(n-1)[2] +b(n-2)[2]]) end: a:= n-> b(n)[1]: seq(a(n), n=1..11);  # Alois P. Heinz, Sep 17 2008

b[0] = {1}; b[1] = {1, 0}; b[n_] := b[n] = Join[b[n-1], b[n-2]]; a[n_] := FromDigits[b[n], 3]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Apr 24 2014 *)

More terms from Jonathan Vos Post, Oct 19 2007

Corrected (a(4) was missing) and extended by Alois P. Heinz, Sep 17 2008

A320986 Fibonacci rabbit sequence number n coded in base four.

Original entry on oeis.org

0, 1, 4, 17, 276, 17681, 18105620, 1186569930001, 79629360058944734484, 350213629188686097286494407640337, 103364819020299355365011185381334444307580296786887956

Offset: 0

Alois P. Heinz, Oct 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..16

Column k=4 of A144287.

b:= proc(n) option remember; `if`(n<2, [n$2], [b(n-1)[1]*
      4^b(n-1)[2]+b(n-2)[1], b(n-1)[2]+b(n-2)[2]])
    end:
a:= n-> b(n)[1]:
seq(a(n), n=0..11);

A320987 Fibonacci rabbit sequence number n coded in base five.

Original entry on oeis.org

0, 1, 5, 26, 655, 81901, 255941280, 99977062581901, 122042312722047375081905, 58194309578918159047081570466564535026, 33873546390630164560327289199086683235486077084641297863363155

Offset: 0

Alois P. Heinz, Oct 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15

Column k=5 of A144287.

b:= proc(n) option remember; `if`(n<2, [n$2], [b(n-1)[1]*
      5^b(n-1)[2]+b(n-2)[1], b(n-1)[2]+b(n-2)[2]])
    end:
a:= n-> b(n)[1]:
seq(a(n), n=0..11);

A320988 Fibonacci rabbit sequence number n coded in base six.

Original entry on oeis.org

0, 1, 6, 37, 1338, 289045, 2247615258, 3775130549469973, 49305824977081270580396826, 1081619448805341465793189347926225612555029, 309896735146874396575135322682564794037430646677718800894678718858010

Offset: 0

Alois P. Heinz, Oct 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15

Column k=6 of A144287.

b:= proc(n) option remember; `if`(n<2, [n$2], [b(n-1)[1]*
      6^b(n-1)[2]+b(n-2)[1], b(n-1)[2]+b(n-2)[2]])
    end:
a:= n-> b(n)[1]:
seq(a(n), n=0..11);

A320989 Fibonacci rabbit sequence number n coded in base seven.

Original entry on oeis.org

0, 1, 7, 50, 2457, 842801, 14164958864, 81658169024988865, 7911779188478711198209076919, 4419091543264985265125879423229594334610523298, 239147782774557159658813554446871692442583155391399547503168277134702036921

Offset: 0

Alois P. Heinz, Oct 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15

Column k=7 of A144287.

b:= proc(n) option remember; `if`(n<2, [n$2], [b(n-1)[1]*
      7^b(n-1)[2]+b(n-2)[1], b(n-1)[2]+b(n-2)[2]])
    end:
a:= n-> b(n)[1]:
seq(a(n), n=0..11);

A320990 Fibonacci rabbit sequence number n coded in base eight.

Original entry on oeis.org

0, 1, 8, 65, 4168, 2134081, 69929570376, 1173223506987487297, 644986443956439734064225751112, 5948949931338226254069168911286743617840154185793, 30164759804754346385078845671118798583946045417223576862481934288047850387443784

Offset: 0

Alois P. Heinz, Oct 25 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..15

Column k=8 of A144287.

b:= proc(n) option remember; `if`(n<2, [n$2], [b(n-1)[1]*
      8^b(n-1)[2]+b(n-2)[1], b(n-1)[2]+b(n-2)[2]])
    end:
a:= n-> b(n)[1]:
seq(a(n), n=0..11);

cp's OEIS Frontend

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

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

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A036299 Binary Fibonacci (or rabbit) sequence.

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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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A005205 Coding Fibonacci numbers.

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A320986 Fibonacci rabbit sequence number n coded in base four.

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A320987 Fibonacci rabbit sequence number n coded in base five.

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