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A144117 Number of Fibonacci parts in the last section of the set of partitions of n.

%I A144117 #14 Aug 06 2015 07:46:30
%S A144117 1,2,3,5,8,13,17,28,37,55,72,104,135,187,243,327,419,557,705,922,1163,
%T A144117 1494,1871,2383,2960,3730,4611,5754,7073,8766,10710,13180,16036,19600,
%U A144117 23736,28859,34788,42075,50529,60811,72747,87184,103907,124019,147330
%N A144117 Number of Fibonacci parts in the last section of the set of partitions of n.
%C A144117 First differences of A144115.
%C A144117 Also number of Fibonacci parts in the n-th section of the set of partitions of any positive integer >= n. - _Omar E. Pol_, Jul 30 2015
%H A144117 Alois P. Heinz, <a href="/A144117/b144117.txt">Table of n, a(n) for n = 1..1000</a>
%F A144117 a(n) = A144115(n) - A144115(n-1).
%p A144117 b:= proc(n) option remember; false end: l:= [0, 1]: for k to 100 do b(l[1]):= true; l:= [l[2], l[1]+l[2]] od: aa:= proc(n, i) option remember; local g, h; if n=0 then [1, 0] elif i=0 or n<0 then [0, 0] else g:= aa(n, i-1); h:= aa(n-i, i); [g[1]+h[1], g[2]+h[2] +`if`(b(i), h[1], 0)] fi end: a:= n-> aa(n, n)[2] -aa(n-1, n-1)[2]: seq(a(n), n=1..60); # _Alois P. Heinz_, Jul 28 2009
%t A144117 Clear[b]; b[_] = False; l = {0, 1}; For[k = 1, k <= 100, k++, b[l[[1]]] = True; l = {l[[2]], l[[1]] + l[[2]]}]; aa[n_, i_] := aa[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i == 0 || n < 0, {0, 0}, g = aa[n, i - 1]; h = aa[n - i, i]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + If[b[i], h[[1]], 0]}]]] ; a[n_] := aa[n, n][[2]] - aa[n - 1, n - 1][[2]]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Jul 30 2015, after _Alois P. Heinz_ *)
%Y A144117 Cf. A000045, A135010, A138121, A138137, A144115, A144116, A144118, A144120.
%K A144117 nonn
%O A144117 1,2
%A A144117 _Omar E. Pol_, Sep 11 2008
%E A144117 More terms from _Alois P. Heinz_, Jul 28 2009