A161027 - OEIS

%I #18 Nov 30 2023 14:06:15

%S 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,

%T 43,48,59,58,67,78,81,92,105,109,123,140,148,160,182,193,214,238,249,

%U 275,305,322,353,386,413,447,490,520,561,611,650,701,762,804,868,938,997,1067,1147

%N Number of partitions of n into Fibonacci numbers where every part appears at least 3 times.

%H Alois P. Heinz, <a href="/A161027/b161027.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1..1000 from R. H. Hardin)

%F 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

%e a(10) = 3 because we have 22222, 2221111, and 1^(10). - _Emeric Deutsch_, Jun 23 2009

%p 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

%p # second Maple program:

%p F:= proc(n, i) option remember; (<<0|1>, <1|1>>^n)[1, 2] end:

%p b:= proc(n, i) option remember; `if`(n=0, 1, (f-> `if`(3*f<=n,

%p       add(b(n-j*f, i+1), j=[0, $3..n/f]), 0))(F(i)))

%p     end:

%p a:= n-> b(n, 2):

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, Feb 23 2019

%t 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]]];

%t a[n_] := b[n, 2];

%t Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Nov 30 2023, after _Alois P. Heinz_ *)

%Y Cf. A000045.

%K nonn

%O 0,7

%A _R. H. Hardin_, Jun 02 2009

%E a(0)=1 prepended by _Alois P. Heinz_, Feb 23 2019