cp's OEIS Frontend

Empirical: n^2*a(n) = (3*n^2-3*n+1)*a(n-1) + 13*(n-1)^2*a(n-2) - 15*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

a(n) appears to be the constant term of (1 + X + 1/X + Y + 1/Y)^n, which has o.g.f. hypergeom([1/2, 1/2],[1],16*x^2/(1-x)^2)/(1-x). - Mark van Hoeij, May 07 2013

From Pierre-Louis Giscard, Jun 25 2014 : (Start)

a(n) is exactly the constant term of (1 + X + 1/X + Y + 1/Y)^n since this generates closed walks on the square lattice with self-loops. Non-constant terms generate walks to the neighbors of a vertex. Removing the 1 is equivalent to removing the self-loops.

a(n) = 3F2([1/2, 1/2 - n/2, -n/2], [1, 1], 16).

a(n) = Sum_{k=0..n} C(n,2k)*C(2k,k)^2.

O.g.f.: 2F1([1/2, 1/2], [1], 16*x^2/(1-x)^2)/(1-x) with 2F1 the Hypergeometric function.

E.g.f.: e^x I_{0}(2x)^2 with I_a(x) the modified Bessel function I of the first kind. (End)

O.g.f.: 1 / AGM(1+3*x, 1-5*x), given a(0)=1, where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Aug 31 2014

a(n) ~ 5^(n+1)/(4*Pi*n). - Vaclav Kotesovec, Oct 03 2016

Empirical: a(n) = 4*n^3 + 14*n + 1.

Conjectures from Colin Barker, May 25 2018: (Start)

G.f.: x*(19 - 15*x + 21*x^2 - x^3) / (1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.

(End)

Empirical: a(n) = 20*n^3 + 30*n + 1.

Conjectures from Colin Barker, May 25 2018: (Start)

G.f.: x*(51 + 17*x + 53*x^2 - x^3) / (1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.

(End)

Empirical: a(n) = 11*n^5 + 65*n^3 + 64*n + 1.

Conjectures from Colin Barker, May 25 2018: (Start)

G.f.: x*(141 + 155*x + 730*x^2 + 150*x^3 + 145*x^4 - x^5) / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

(End)

Empirical: a(n) = 77*n^5 + 175*n^3 + 140*n + 1.

Conjectures from Colin Barker, May 25 2018: (Start)

G.f.: x*(393 + 1787*x + 4882*x^2 + 1782*x^3 + 397*x^4 - x^5) / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.

(End)