A292341 Number of unrooted loops of length 2n on the square lattice that have winding number +1 around a fixed off-lattice point.

1, 16, 232, 3328, 47957, 696304, 10187288, 150087168, 2224889247, 33160970672, 496608054904, 7468314975488, 112731489535747, 1707278435651920, 25932766975385096, 394956591009678336, 6029683178394959854, 92254556123206383072

Offset

2,2

Comments

Here a rooted loop on the square lattice of length 2n is a sequence in Z^2 of length 2n such that (cyclically) consecutive pairs of points have distance 1. An unrooted loop is a rooted loop modulo cyclic permutations.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 2..800

T. Budd, Winding of simple walks on the square lattice, arXiv:1709.04042 [math.CO], 2017.

Formula

G.f.: A(x) = q^2/(1-q^4) with q=q(16x) the Jacobi nome of parameter m=16x.

Example

For n=2 there is a(2)=1 such loop: the contour of the unit square (in counterclockwise direction).

Mathematica

a[n_] := SeriesCoefficient[q^2/(1-q^4) /. q->EllipticNomeQ[16 x], {x, 0, n}]

Crossrefs

Cf. A005797.

Sequence in context: A230234 A274467 A119463 * A222389 A222938 A265469

Adjacent sequences: A292338 A292339 A292340 * A292342 A292343 A292344

Keyword

nonn,walk

Author

Timothy Budd, Sep 14 2017

Status

approved