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%I #37 Feb 21 2017 02:38:12
%S 0,1,4,9,16,31,52,80,133,197,298,428,621,879,1230,1696,2329,3142,4231,
%T 5619,7447,9781,12771,16553,21391,27440,35089,44600,56510,71232,89538,
%U 112011,139759,173679,215279,265840,327527,402162,492703,601830,733550,891634
%N Total sum of Fibonacci parts in all partitions of n.
%H Alois P. Heinz, <a href="/A199936/b199936.txt">Table of n, a(n) for n = 0..5000</a>
%F G.f.: Sum_{i>=2} Fibonacci(i)*x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Feb 01 2017
%e For n = 6 we have:
%e --------------------------------------
%e . Sum of
%e Partitions Fibonacci parts
%e --------------------------------------
%e 6 .......................... 0
%e 3 + 3 ...................... 6
%e 4 + 2 ...................... 2
%e 2 + 2 + 2 .................. 6
%e 5 + 1 ...................... 6
%e 3 + 2 + 1 .................. 6
%e 4 + 1 + 1 .................. 2
%e 2 + 2 + 1 + 1 .............. 6
%e 3 + 1 + 1 + 1 .............. 6
%e 2 + 1 + 1 + 1 + 1 .......... 6
%e 1 + 1 + 1 + 1 + 1 + 1 ...... 6
%e ------------------------------------
%e Total ..................... 52
%e So a(6) = 52.
%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p `if`(i>n, 0, ((p, m)-> p +`if`(issqr(m+4) or issqr(m-4),
%p [0, p[1]*i], 0))(b(n-i, i), 5*i^2)) +b(n, i-1)))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Feb 01 2017
%t max = 42; F = Fibonacci; gf = Sum[F[i]*x^F[i]/(1-x^F[i]), {i, 2, max}] / Product[1-x^j, {j, 1, max}] + O[x]^max; CoefficientList[gf, x] (* _Jean-François Alcover_, Feb 21 2017, after _Ilya Gutkovskiy_ *)
%t b[n_, i_] := b[n, i] = If[n==0, {1, 0}, If[i<1, 0, If[i>n, 0, Function[{p, m}, p+If[IntegerQ @ Sqrt[m+4] || IntegerQ @ Sqrt[m-4], {0, p[[1]]*i}, 0] ][b[n-i, i], 5*i^2]]+b[n, i-1]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 21 2017, after _Alois P. Heinz_ *)
%Y Cf. A000045, A066186, A073118, A144115, A194544, A194545.
%K nonn
%O 0,3
%A _Omar E. Pol_, Nov 21 2011
%E More terms from _Alois P. Heinz_, Nov 21 2011
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