OFFSET
0,3
COMMENTS
Also number of n-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 4).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500 (terms 0..300 from Vaclav Kotesovec)
Vaclav Kotesovec, Recurrence of order 8 (conjectured)
Wikipedia, Taxicab geometry.
FORMULA
Conjecture: a(n) ~ 2 * 16^n / (11*Pi*n). - Vaclav Kotesovec, Nov 04 2019
PROG
(PARI) {a(n) = polcoef(polcoef((sum(k=-2, 2, x^k)*sum(k=-2, 2, y^k)-(x+1+1/x)*(y+1+1/y))^n, 0), 0)}
(PARI) {a(n) = polcoef(polcoef((sum(k=0, 4, (x^k+1/x^k)*(y^(4-k)+1/y^(4-k)))-x^4-1/x^4-y^4-1/y^4)^n, 0), 0)}
(PARI) f(n) = (x^(n+1)-1/x^n)/(x-1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(2)^k*f(1)^(n-k), 0)^2)
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Seiichi Manyama, Nov 03 2019
STATUS
approved