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%I M1539 #63 Sep 20 2020 03:17:16
%S 0,1,2,5,22,181,5814,1488565,12194330294,25573364166211253,
%T 439347050970302571643057846,
%U 15829145720289447797800874537321282579904181,9797766637414564027586288536574448245991597197836000123235901011048118
%N Fibonacci numbers (or rabbit sequence) converted to decimal.
%C 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
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005203/b005203.txt">Table of n, a(n) for n = 0..17</a>
%H J. L. Davison, <a href="http://dx.doi.org/10.1090/S0002-9939-1977-0429778-5">A series and its associated continued fraction</a>, Proc. Amer. Math. Soc., 63 (1977), 29-32.
%H H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/15-4/gould.pdf">Sequences associated with t-ary coding of Fibonacci's rabbits</a>, Fib. Quart., 15 (1977), 311-318.
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrab.html">The Fibonacci Rabbit Sequence</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibrep.html">Rabbit Sequence in Zeckendorf Expansion (A003714)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RabbitSequence.html">Rabbit Sequence</a>
%F a(0) = 0, a(1) = 1, a(n) = a(n-1) * 2^F(n-1) + a(n-2).
%F a(n) = rewrite_0to1_1to10_n_i_times(0, n) [ Each 0->1, 1->10 in binary expansion ]
%p 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;
%p 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;
%t 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 *)
%Y Cf. A000045, A048707, A003714, A048721, A048722, A048678, A048679, A048680, A005205, A063896.
%Y Column k=2 of A144287.
%K nonn,base
%O 0,3
%A _N. J. A. Sloane_
%E Comments and more terms from _Antti Karttunen_, Mar 30 1999
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