A268128 a(n) = (A000123(n) - A001316)/2.

0, 0, 1, 1, 4, 5, 8, 9, 17, 21, 28, 33, 45, 53, 66, 75, 100, 117, 140, 161, 193, 221, 258, 291, 344, 389, 446, 499, 573, 639, 722, 797, 913, 1013, 1132, 1249, 1393, 1533, 1698, 1859, 2060, 2253, 2478, 2699, 2965, 3223, 3522, 3813, 4173, 4517, 4910, 5299, 5753

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, A268127.

Programs

Mathematica

b[0] = 1; b[n_] := b[n] = b[Floor[n/2]] + b[n - 1];
c[n_] := Sum[Mod[Binomial[n, k], 2], {k, 0, n}];
a[n_] := (b[n] - c[n])/2;
Table[a[n], {n, 0, 52}] (* 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/2)])
    return A[n]
[(b(n)-prod(x+1 for x in n.digits(2)))/2 for n in [0..60]]

Formula

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/2)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*2^i is the binary representation of n. Then a(n) = (1/2)*(b(n) - c(n)).