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URL: https://oeis.org/A161027

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A161027
Number of partitions of n into Fibonacci numbers where every part appears at least 3 times.
1
1, 0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 6, 5, 6, 10, 8, 9, 14, 13, 16, 20, 19, 23, 30, 30, 33, 41, 43, 48, 59, 58, 67, 78, 81, 92, 105, 109, 123, 140, 148, 160, 182, 193, 214, 238, 249, 275, 305, 322, 353, 386, 413, 447, 490, 520, 561, 611, 650, 701, 762, 804, 868, 938, 997, 1067, 1147
OFFSET
0,7
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..1000 from R. H. Hardin)
FORMULA
G.f.: Product_{j>=2} (1 + x^(3*F(j))/(1 - x^(F(j)))), where F = A000045 are the Fibonacci numbers. - Emeric Deutsch, Jun 23 2009
EXAMPLE
a(10) = 3 because we have 22222, 2221111, and 1^(10). - Emeric Deutsch, Jun 23 2009
MAPLE
with(combinat): g := product(1+x^(3*fibonacci(j))/(1-x^fibonacci(j)), j = 2 .. 10): gser := series(g, x = 0, 95): seq(coeff(gser, x, n), n = 0 .. 71); # Emeric Deutsch, Jun 23 2009
# Alternative:
F:= proc(n, i) option remember; (<<0|1>, <1|1>>^n)[1, 2] end:
b:= proc(n, i) option remember; `if`(n=0, 1, (f-> `if`(3*f<=n,
add(b(n-j*f, i+1), j=[0, $3..n/f]), 0))(F(i)))
end:
a:= n-> b(n, 2):
seq(a(n), n=0..80); # Alois P. Heinz, Feb 23 2019
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == 0, 1, With[{f = Fibonacci[i]}, If[3*f <= n, Sum[b[n - j*f, i + 1], {j, Prepend[Range[3, n/f], 0]}], 0]]];
a[n_] := b[n, 2];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 30 2023, after Alois P. Heinz *)
CROSSREFS
Cf. A000045.
Sequence in context: A306878 A161308 A161242 * A161078 A161294 A161269
KEYWORD
nonn
AUTHOR
R. H. Hardin, Jun 02 2009
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Feb 23 2019
STATUS
approved