A144115 Total number of Fibonacci parts in all partitions of n.
1, 3, 6, 11, 19, 32, 49, 77, 114, 169, 241, 345, 480, 667, 910, 1237, 1656, 2213, 2918, 3840, 5003, 6497, 8368, 10751, 13711, 17441, 22052, 27806, 34879, 43645, 54355, 67535, 83571, 103171, 126907, 155766, 190554, 232629, 283158, 343969, 416716, 503900, 607807
Offset: 1
Keywords
Examples
From _Omar E. Pol_, Nov 20 2011 (Start): For n = 6 we have: -------------------------------------- . Number of Partitions Fibonacci parts -------------------------------------- 6 .......................... 0 3 + 3 ...................... 2 4 + 2 ...................... 1 2 + 2 + 2 .................. 3 5 + 1 ...................... 2 3 + 2 + 1 .................. 3 4 + 1 + 1 .................. 2 2 + 2 + 1 + 1 .............. 4 3 + 1 + 1 + 1 .............. 4 2 + 1 + 1 + 1 + 1 .......... 5 1 + 1 + 1 + 1 + 1 + 1 ...... 6 ------------------------------------ Total ..................... 32 So a(6) = 32. (End)
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..8000
Programs
-
Maple
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, b(n, i-1)+ (p-> p+`if`((t-> issqr(t+4) or issqr(t-4) )(5*i^2), [0, p[1]], 0))(b(n-i, min(n-i, i))))) end: a:= n-> b(n$2)[2]: seq(a(n), n=1..60); # Alois P. Heinz, Jun 24 2009, revised Aug 06 2019 -
Mathematica
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 *)
Formula
G.f.: Sum_{i>=2} x^Fibonacci(i)/(1 - x^Fibonacci(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Jan 24 2017
Extensions
More terms from Alois P. Heinz, Jun 24 2009
Comments