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A199936
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Total sum of Fibonacci parts in all partitions of n.
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4
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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
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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]:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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