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

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

Offset: 0

Author

Amarnath Murthy, Apr 20 2001

Keywords

• base
• nonn
• easy

Comments

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.

Examples

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

References

• 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.

Links

• 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.

Crossrefs

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

Column k=10 of A144287.

Programs

Maple

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

Mathematica

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

PARI

{ 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

Formula

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]

Extensions

More terms from Hieronymus Fischer, Jun 26 2007