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%I A076739 #36 Oct 30 2023 08:01:19
%S A076739 1,1,2,4,7,14,26,49,94,177,336,637,1206,2288,4335,8216,15574,29515,
%T A076739 55943,106030,200959,380889,721906,1368251,2593291,4915135,9315811,
%U A076739 17656534,33464955,63427148,120215370,227847814,431846824,818492263
%N A076739 Number of compositions of n into Fibonacci numbers (1 counted as single Fibonacci number).
%C A076739 From _Gary W. Adamson_, Sep 12 2008: (Start)
%C A076739 Equals right border of triangle A144172 and row sums with offset 1.
%C A076739 Equals INVERT transform of the characteristic function of the Fibonacci numbers starting with offset 1: (1, 1, 1, 0, 1, ...) (if the first "1" is retained: = 1, 1, 2, 4, 7, 14, ...). (End)
%D A076739 A. Knopfmacher & N. Robbins, On binary and Fibonacci compositions, Annales Univ. Sci. Budapest, Sect. Comp. 22 (2003) 193-206. - _Neville Robbins_, Mar 06 2010
%H A076739 Alois P. Heinz, <a href="/A076739/b076739.txt">Table of n, a(n) for n = 0..3600</a> (first 301 terms from T. D. Noe)
%F A076739 G.f.: 1/(1-Sum_{k>1} x^Fibonacci(k)). - _Vladeta Jovovic_, Jun 20 2003
%F A076739 a(n) ~ c * d^n, where d=1.8953300920998046150867311236880760382884608526935119695..., c=0.5615834114640436146286049301387868479914202616794427372... - _Vaclav Kotesovec_, May 01 2014
%e A076739 a(4) = 7 since 3+1 = 2+2 = 2+1+1 = 1+3 = 1+2+1 = 1+1+2 = 1+1+1+1.
%p A076739 a:= proc(n) option remember; local r, f;
%p A076739       if n=0 then 1 else r, f:= 0, [1$2];
%p A076739         while f[2] <= n do r:= r+a(n-f[2]);
%p A076739           f:= [f[2], f[1]+f[2]]
%p A076739         od; r
%p A076739       fi
%p A076739     end:
%p A076739 seq(a(n), n=0..35);  # _Alois P. Heinz_, Feb 20 2017
%t A076739 max=40; 1/(1-Total[x^Fibonacci[Range[2, Ceiling[Sqrt[max]]+2]]]) + O[x]^max // CoefficientList[#, x]& (* _Jean-François Alcover_, Mar 29 2017, after _Vladeta Jovovic_ *)
%Y A076739 Cf. A080888.
%Y A076739 Cf. A144172, A010056. - _Gary W. Adamson_, Sep 12 2008
%K A076739 nonn
%O A076739 0,3
%A A076739 _David W. Wilson_, Jun 19 2003

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