A056463 Number of primitive (aperiodic) palindromes using exactly two different symbols.

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

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

References

• 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]

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Column 2 of A327873.

Cf. A056453, A056458.

Programs

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

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

Formula

a(n) = Sum_{d|n} mu(d)*A056453(n/d).

G.f.: Sum_{k>=1} mu(k)*2*x^(3*k)/((1 - 2*x^(2*k))*(1 - x^k)). - Andrew Howroyd, Sep 29 2019

Extensions

Terms a(32) and beyond from Andrew Howroyd, Sep 28 2019