A199936 Total sum of Fibonacci parts in all partitions of n.
0, 1, 4, 9, 16, 31, 52, 80, 133, 197, 298, 428, 621, 879, 1230, 1696, 2329, 3142, 4231, 5619, 7447, 9781, 12771, 16553, 21391, 27440, 35089, 44600, 56510, 71232, 89538, 112011, 139759, 173679, 215279, 265840, 327527, 402162, 492703, 601830, 733550, 891634
Offset: 0
Keywords
Examples
For n = 6 we have: -------------------------------------- . Sum of Partitions Fibonacci parts -------------------------------------- 6 .......................... 0 3 + 3 ...................... 6 4 + 2 ...................... 2 2 + 2 + 2 .................. 6 5 + 1 ...................... 6 3 + 2 + 1 .................. 6 4 + 1 + 1 .................. 2 2 + 2 + 1 + 1 .............. 6 3 + 1 + 1 + 1 .............. 6 2 + 1 + 1 + 1 + 1 .......... 6 1 + 1 + 1 + 1 + 1 + 1 ...... 6 ------------------------------------ Total ..................... 52 So a(6) = 52.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..5000
Programs
-
Maple
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, `if`(i>n, 0, ((p, m)-> p +`if`(issqr(m+4) or issqr(m-4), [0, p[1]*i], 0))(b(n-i, i), 5*i^2)) +b(n, i-1))) end: a:= n-> b(n$2)[2]: seq(a(n), n=0..50); # Alois P. Heinz, Feb 01 2017 -
Mathematica
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 *) 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 *)
Formula
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
Extensions
More terms from Alois P. Heinz, Nov 21 2011