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A144115 Total number of Fibonacci parts in all partitions of n.

%I A144115 #33 Aug 07 2019 15:32:33
%S A144115 1,3,6,11,19,32,49,77,114,169,241,345,480,667,910,1237,1656,2213,2918,
%T A144115 3840,5003,6497,8368,10751,13711,17441,22052,27806,34879,43645,54355,
%U A144115 67535,83571,103171,126907,155766,190554,232629,283158,343969,416716,503900,607807
%N A144115 Total number of Fibonacci parts in all partitions of n.
%C A144115 a(n) is also the sum of the differences between the sum of f-th largest and the sum of (f+1)-st largest elements in all partitions of n for all Fibonacci parts f. - _Omar E. Pol_, Oct 27 2012
%H A144115 Alois P. Heinz, <a href="/A144115/b144115.txt">Table of n, a(n) for n = 1..8000</a>
%F A144115 G.f.: Sum_{i>=2} x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Jan 24 2017
%e A144115 From _Omar E. Pol_, Nov 20 2011 (Start):
%e A144115 For n = 6 we have:
%e A144115 --------------------------------------
%e A144115 .                        Number of
%e A144115 Partitions            Fibonacci parts
%e A144115 --------------------------------------
%e A144115 6 .......................... 0
%e A144115 3 + 3 ...................... 2
%e A144115 4 + 2 ...................... 1
%e A144115 2 + 2 + 2 .................. 3
%e A144115 5 + 1 ...................... 2
%e A144115 3 + 2 + 1 .................. 3
%e A144115 4 + 1 + 1 .................. 2
%e A144115 2 + 2 + 1 + 1 .............. 4
%e A144115 3 + 1 + 1 + 1 .............. 4
%e A144115 2 + 1 + 1 + 1 + 1 .......... 5
%e A144115 1 + 1 + 1 + 1 + 1 + 1 ...... 6
%e A144115 ------------------------------------
%e A144115 Total ..................... 32
%e A144115 So a(6) = 32. (End)
%p A144115 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
%p A144115       b(n, i-1)+ (p-> p+`if`((t-> issqr(t+4) or issqr(t-4)
%p A144115       )(5*i^2), [0, p[1]], 0))(b(n-i, min(n-i, i)))))
%p A144115     end:
%p A144115 a:= n-> b(n$2)[2]:
%p A144115 seq(a(n), n=1..60); # _Alois P. Heinz_, Jun 24 2009, revised Aug 06 2019
%t A144115 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]]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Jul 30 2015, after _Alois P. Heinz_ *)
%Y A144115 Cf. A000045, A006128, A037032, A144116, A144117, A144118, A199936, A309537.
%K A144115 nonn
%O A144115 1,2
%A A144115 _Omar E. Pol_, Sep 11 2008
%E A144115 More terms from _Alois P. Heinz_, Jun 24 2009