OFFSET
0,11
COMMENTS
A combinatorial interpretation is given in the Edgar link.
LINKS
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, Characterizing the number of m-ary partitions modulo m.
Tom Edgar, The distribution of the number of parts of m-ary partitions modulo m., arXiv:1603.00085 [math.CO], 2016.
FORMULA
Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/5)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*5^i is the base 5 representation of n. Then a(n) = (1/5)*(b(n) - c(n)).
MATHEMATICA
b[0] = 1; b[n_] := b[n] = b[n-1] + b[Floor[n/5]];
c[n_] := If[OddQ[n], 2 Count[Table[Binomial[n, k], {k, 0, (n-1)/2}], c_ /; !Divisible[c, 5]], 2 Count[Table[Binomial[n, k], {k, 0, (n-2)/2}], c_ /; !Divisible[c, 5]] + Boole[!Divisible[Binomial[n, n/2], 5]]];
a[n_] := (b[n] - c[n])/5;
Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Feb 15 2019 *)
PROG
(Sage)
def b(n):
A=[1]
for i in [1..n]:
A.append(A[i-1] + A[i//5])
return A[n]
print([(b(n)-prod(x+1 for x in n.digits(5)))/5 for n in [0..63]])
CROSSREFS
KEYWORD
nonn
AUTHOR
Tom Edgar, Mar 22 2016
STATUS
approved