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%I #16 Dec 18 2023 08:26:09
%S 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,
%T 35,36,37,38,49,52,55,58,64,67,70,73,82,85,88,91,103,106,109,112,130,
%U 136,142,148,158,164,170,176,190,196,202,208,226,232,238,244
%N a(n) = (A005705(n) - A268444(n))/4.
%H G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.9.880">Characterizing the number of m-ary partitions modulo m</a>, The American Mathematical Monthly, Vol. 122, No. 9 (November 2015), pp. 880-885.
%H G. E. Andrews, A. S. Fraenkel, and J. A. Sellers, <a href="https://georgeandrews1.github.io/pdf/300.pdf">Characterizing the number of m-ary partitions modulo m</a>.
%H Tom Edgar, <a href="http://arxiv.org/abs/1603.00085">The distribution of the number of parts of m-ary partitions modulo m.</a>, arXiv:1603.00085 [math.CO], 2016.
%F 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)).
%o (Sage)
%o def b(n):
%o A=[1]
%o for i in [1..n]:
%o A.append(A[i-1] + A[i//4])
%o return A[n]
%o print([(b(n)-prod(x+1 for x in n.digits(4)))/4 for n in [0..63]])
%Y Cf. A005705, A268444, A268127, A268128.
%K nonn
%O 0,9
%A _Tom Edgar_, Feb 04 2016
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