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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.

%I A061107 #29 Apr 11 2025 03:34:02
%S A061107 0,1,10,101,10110,10110101,1011010110110,101101011011010110101,
%T A061107 1011010110110101101011011010110110,
%U A061107 1011010110110101101011011010110110101101011011010110101
%N A061107 a(0) = 0, a(1) = 1, a(n) is the concatenation of a(n-2) and a(n-1) for n > 1.
%C A061107 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.
%D A061107 Amarnath Murthy, Smarandache Reverse auto correlated sequences and some Fibonacci derived sequences, Smarandache Notions Journal Vol. 12, No. 1-2-3, Spring 2001.
%D A061107 Ian Stewart, The Magical Maze.
%H A061107 Harry J. Smith, <a href="/A061107/b061107.txt">Table of n, a(n) for n = 0..15</a>
%H A061107 Duaa Abdullah and Jasem Hamoud, <a href="https://arxiv.org/abs/2504.04087">Dynamic and Programmatic Analysis of Fibonacci Word Density</a>, arXiv:2504.04087 [math.CO], 2025. See p. 10.
%F A061107 a(0) = 0, a(1) =1, a(n) = concatenation of a(n-1) and a(n-2).
%F A061107 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
%F A061107 a(n) = A036299(n-1), n>0. - _R. J. Mathar_, Oct 02 2008
%F A061107 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]
%e A061107 a(0) = 0, a(1) = 1, a(2) = a(1)a(0)= 10, etc.
%p A061107 A[0]:= 0: A[1]:= 1: A[2]:= 10:
%p A061107 for n from 3 to 20 do
%p A061107 A[n]:= 10^(ilog10(A[n-2])+1)*A[n-1]+A[n-2]
%p A061107 od:
%p A061107 seq(A[n],n=0..10); # _Robert Israel_, Apr 30 2015
%t A061107 nxt[{a_,b_}]:={b,FromDigits[Join[IntegerDigits[b],IntegerDigits[a]]]}; Transpose[NestList[nxt,{0,1},10]][[1]] (* _Harvey P. Dale_, Jul 05 2015 *)
%o A061107 (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
%Y A061107 Cf. A063896, A131242. See A005203 for the sequence version converted to decimal.
%Y A061107 Column k=10 of A144287.
%K A061107 base,nonn,easy
%O A061107 0,3
%A A061107 _Amarnath Murthy_, Apr 20 2001
%E A061107 More terms from _Hieronymus Fischer_, Jun 26 2007