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Total number of Fibonacci parts in all partitions of n.
9

%I #33 Aug 07 2019 15:32:33

%S 1,3,6,11,19,32,49,77,114,169,241,345,480,667,910,1237,1656,2213,2918,

%T 3840,5003,6497,8368,10751,13711,17441,22052,27806,34879,43645,54355,

%U 67535,83571,103171,126907,155766,190554,232629,283158,343969,416716,503900,607807

%N Total number of Fibonacci parts in all partitions of n.

%C 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 Alois P. Heinz, <a href="/A144115/b144115.txt">Table of n, a(n) for n = 1..8000</a>

%F G.f.: Sum_{i>=2} x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Jan 24 2017

%e From _Omar E. Pol_, Nov 20 2011 (Start):

%e For n = 6 we have:

%e --------------------------------------

%e . Number of

%e Partitions Fibonacci parts

%e --------------------------------------

%e 6 .......................... 0

%e 3 + 3 ...................... 2

%e 4 + 2 ...................... 1

%e 2 + 2 + 2 .................. 3

%e 5 + 1 ...................... 2

%e 3 + 2 + 1 .................. 3

%e 4 + 1 + 1 .................. 2

%e 2 + 2 + 1 + 1 .............. 4

%e 3 + 1 + 1 + 1 .............. 4

%e 2 + 1 + 1 + 1 + 1 .......... 5

%e 1 + 1 + 1 + 1 + 1 + 1 ...... 6

%e ------------------------------------

%e Total ..................... 32

%e So a(6) = 32. (End)

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,

%p b(n, i-1)+ (p-> p+`if`((t-> issqr(t+4) or issqr(t-4)

%p )(5*i^2), [0, p[1]], 0))(b(n-i, min(n-i, i)))))

%p end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Jun 24 2009, revised Aug 06 2019

%t 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 Cf. A000045, A006128, A037032, A144116, A144117, A144118, A199936, A309537.

%K nonn

%O 1,2

%A _Omar E. Pol_, Sep 11 2008

%E More terms from _Alois P. Heinz_, Jun 24 2009