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A309537 Total number of Fibonacci parts in all compositions of n. 2
0, 1, 3, 8, 19, 46, 106, 241, 541, 1198, 2629, 5724, 12380, 26625, 56978, 121413, 257740, 545308, 1150272, 2419856, 5078336, 10633921, 22222338, 46353669, 96525324, 200686620, 416645184, 863834256, 1788756288, 3699688128, 7643727360, 15776156928, 32529718272 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
G.f.: Sum_{k>=2} x^Fibonacci(k)*(1-x)^2/(1-2*x)^2.
a(n) ~ c * 2^n * n, where c = 0.22756969930196647294851075611776578612085598114... - Vaclav Kotesovec, Aug 18 2019
MAPLE
a:= proc(n) option remember; add(a(n-j)+`if`((t->issqr(t+4)
or issqr(t-4))(5*j^2), ceil(2^(n-j-1)), 0), j=1..n)
end:
seq(a(n), n=0..33);
MATHEMATICA
a[n_] := a[n] = Sum[a[n - j] + With[{t = 5 j^2}, If[IntegerQ@Sqrt[t + 4] || IntegerQ@Sqrt[t - 4], Ceiling[2^(n - j - 1)], 0]], {j, 1, n}];
a /@ Range[0, 33] (* Jean-François Alcover, Dec 29 2020, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A026789 A308372 A096576 * A126874 A284942 A121811
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 06 2019
STATUS
approved

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Last modified February 20 18:38 EST 2024. Contains 370217 sequences. (Running on oeis4.)