A268127 a(n) = (A005704(n)-A006047(n))/3.

0, 0, 0, 1, 1, 1, 3, 3, 3, 7, 8, 9, 12, 13, 14, 19, 20, 21, 30, 33, 36, 42, 45, 48, 57, 60, 63, 79, 86, 93, 103, 111, 119, 132, 141, 150, 168, 180, 192, 209, 222, 235, 257, 271, 285, 316, 335, 354, 380, 400, 420, 453, 474, 495, 543, 573, 603, 639, 672, 705, 747

Offset: 0

Author

Tom Edgar, Jan 26 2016

Keywords

• nonn

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.

Crossrefs

Cf. A005704, A006047, A268128.

Programs

Mathematica

b[n_] := b[n] = If[n <= 2, n+1, b[n-1] + b[Floor[n/3]]];
c = Nest[Join[#, 2#, 3#]&, {1}, 4];
a[n_] := (b[n] - c[[n+1]])/3;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Dec 12 2018 *)

Sage

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[floor(i/3)])
    return A[n]
[(b(n)-prod(x+1 for x in n.digits(3)))/3 for n in [0..60]]

Formula

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/3)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*3^i is the ternary representation of n. Then a(n) = (1/3)*(b(n) - c(n)).