OFFSET
0,2
COMMENTS
In the preprint and the paper by Karpov et al., a(n) (resp. 2*a(n)) is the size of the Condorcet domain on 2*n (resp. 2*n+1) alternatives defined by the so-called even 1N33N1 scheme, cf. A144685. - Andrey Zabolotskiy, Jan 27 2024
LINKS
Robert Israel, Table of n, a(n) for n = 0..1461
J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, eq. (30) and (32).
Alexander Karpov, Klas Markström, Søren Riis and Bei Zhou, Bipartite peak-pit domains, arXiv:2308.02817 [cs.DM], 2023-2024.
Alexander Karpov, Klas Markström, Søren Riis, and Bei Zhou, Coherent domains and improved lower bounds for the maximum size of Condorcet domains, Discrete Applied Mathematics, Volume 370, Pages 57-70 (2025). See pp. 68-69.
FORMULA
Conjecture: n*a(n) + 2*(-4*n+3)*a(n-1) + 12*(n-2)*a(n-2) + 8*(2*n-3)*a(n-3) = 0.
From Robert Israel, Mar 31 2019: (Start)
Conjecture verified (for n >= 4) using the differential equation (16*x^3 + 12*x^2 - 8*x + 1)*y' + (24*x^2 - 2)*y -12*x^2 + 2*x = 0 satisfied by the g.f.
a(n) ~ (sqrt(2)/4)*(2 + 2*sqrt(2))^n. (End)
a(n) = (1/n) * Sum_{k=1..n} k * Pell(k+1) * binomial(2*n,n-k) for n > 0. - Seiichi Manyama, Dec 10 2025
MAPLE
f:= gfun:-rectoproc({n*a(n) +2*(-4*n+3)*a(n-1) +12*(n-2)*a(n-2) +8*(2*n-3)*a(n-3), a(0)=1, a(1)=2, a(2)=9, a(3)=42}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 31 2019
MATHEMATICA
CoefficientList[2 x^2/(4 x^2 + 2x + Sqrt[1 - 4x] - 1) + O[x]^25, x] (* Jean-François Alcover, Aug 26 2022 *)
PROG
(SageMath) (2*x^2/(4*x^2+2*x+sqrt(1-4*x)-1)).series(x, 25).coefficients(x, sparse=False) # Stefano Spezia, Mar 19 2025
(PARI) pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n) = if(n==0, 1, sum(k=1, n, k*pell(k+1)*binomial(2*n, n-k))/n); \\ Seiichi Manyama, Dec 10 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Jul 09 2017
STATUS
approved