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A199936 Total sum of Fibonacci parts in all partitions of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified April 4 08:58 EDT 2020. Contains 333213 sequences. (Running on oeis4.)