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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

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

Example

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.

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 *)

Crossrefs

Cf. A000045, A066186, A073118, A144115, A194544, A194545.

Sequence in context: A073141 A093175 A138992 * A326958 A281904 A007679

Adjacent sequences: A199933 A199934 A199935 * A199937 A199938 A199939

Keyword

nonn

Author

Omar E. Pol, Nov 21 2011

Extensions

More terms from Alois P. Heinz, Nov 21 2011

Status

approved