OFFSET
0,2
COMMENTS
Also number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 5).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..300 (terms 0..100 from Vaclav Kotesovec)
Wikipedia, Taxicab geometry.
FORMULA
Conjecture: a(n) ~ 400^n / (17*Pi*n). - Vaclav Kotesovec, Nov 04 2019
PROG
(PARI) {a(n) = polcoef(polcoef(((x^5+x^3+x+1/x+1/x^3+1/x^5)*(y^5+y^3+y+1/y+1/y^3+1/y^5)-(x^3+x+1/x+1/x^3)*(y^3+y+1/y+1/y^3))^(2*n), 0), 0)}
(PARI) {a(n) = polcoef(polcoef((sum(k=0, 5, (x^k+1/x^k)*(y^(5-k)+1/y^(5-k)))-x^5-1/x^5-y^5-1/y^5)^(2*n), 0), 0)}
(PARI) f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
a(n) = sum(k=0, 2*n, (-1)^k*binomial(2*n, k)*polcoef(f(2)^k*f(1)^(2*n-k), 0)^2)
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
Seiichi Manyama, Nov 03 2019
STATUS
approved