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(Python)
from sympy import mobius, divisors
def A056463(n): return sum(mobius(n//d)*((1<<(d+1>>1))-2) for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 18 2024
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G.f.: Sum_{k>=1} mu(k)*2*x^(3*k)/((1 - 2*x^(2*k))*(1 - x^k)). - Andrew Howroyd, Sep 29 2019
(PARI) seq(n)={Vec(sum(k=1, n\3, moebius(k)*2*x^(3*k)/((1 - 2*x^(2*k))*(1 - x^k)) + O(x*x^n)), -n)} \\ Andrew Howroyd, Sep 29 2019
0, 0, 2, 2, 6, 4, 14, 12, 28, 24, 62, 54, 126, 112, 246, 240, 510, 476, 1022, 990, 2030, 1984, 4094, 4020, 8184, 8064, 16352, 16254, 32766, 32484, 65534, 65280, 131006, 130560, 262122, 261576, 524286, 523264, 1048446, 1047540, 2097150, 2094988, 4194302, 4192254
Andrew Howroyd, <a href="/A056463/b056463.txt">Table of n, a(n) for n = 1..1000</a>
a(n) = Sum _{d|n} mu(d)*A056453(n/d) where d divides n.
Terms a(32) and beyond from Andrew Howroyd, Sep 28 2019
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Marks R. Nester (nesterm(AT)dpi.qld.gov.au)
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]