A268443 a(n) = (A005705(n) - A268444(n))/4.

0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 3, 6, 6, 6, 6, 11, 12, 13, 14, 17, 18, 19, 20, 25, 26, 27, 28, 35, 36, 37, 38, 49, 52, 55, 58, 64, 67, 70, 73, 82, 85, 88, 91, 103, 106, 109, 112, 130, 136, 142, 148, 158, 164, 170, 176, 190, 196, 202, 208, 226, 232, 238, 244

Offset: 0

Author

Tom Edgar, Feb 04 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. A005705, A268444, A268127, A268128.

Programs

Sage

def b(n):
    A=[1]
    for i in [1..n]:
        A.append(A[i-1] + A[i//4])
    return A[n]
print([(b(n)-prod(x+1 for x in n.digits(4)))/4 for n in [0..63]])

Formula

Let b(0) = 1 and b(n) = b(n-1) + b(floor(n/4)) and let c(n) = Product_{i=0..k}(n_i+1) where n = Sum_{i=0..k}n_i*4^i is the base 4 representation of n. Then a(n) = (1/4)*(b(n) - c(n)).