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A005205 Coding Fibonacci numbers.

%I A005205 M2877 #36 Jan 05 2025 19:51:33
%S A005205 1,3,10,93,2521,612696,4019900977,6409020585966267,
%T A005205 67040619014505181883304178,
%U A005205 1118048584563024433220786501983631190591549,195042693446883195450571898296824337898272003171567594807867055549521
%N A005205 Coding Fibonacci numbers.
%C A005205 Binary Fibonacci (or rabbit) sequence A036299, read in base 3, then converted to decimal. - _Jonathan Vos Post_, Oct 19 2007
%D A005205 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005205 Alois P. Heinz, <a href="/A005205/b005205.txt">Table of n, a(n) for n = 1..16</a>
%H A005205 H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://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.
%e A005205 a(0) = 1 because A036299(0) = "1" and 1 base 3 = 1 base 10.
%e A005205 a(1) = 3 because A036299(1) = "10" and 10 base 3 = 3 base 10.
%e A005205 a(2) = 10 because A036299(2) = "101" and 101 base 3 = 10 base 10.
%e A005205 a(3) = 93 because A036299(3) = "10110" and 10110 base 3 = 93 base 10.
%e A005205 a(4) = 2521 because A036299(4) = "10110101" and 10110101 base 3 = 2521 base 10.
%e A005205 a(5) = 612696 because A036299(5) = "1011010110110" and 1011010110110 base 3 = 612696 base 10.
%p A005205 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
%t A005205 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 *)
%Y A005205 Cf. A005203, A036299.
%Y A005205 Column k=3 of A144287.
%K A005205 nonn,base
%O A005205 1,2
%A A005205 _N. J. A. Sloane_
%E A005205 More terms from _Jonathan Vos Post_, Oct 19 2007
%E A005205 Corrected (a(4) was missing) and extended by _Alois P. Heinz_, Sep 17 2008