Timothy Budd - cp's OEIS frontend

User: Timothy Budd

Timothy Budd has authored 2 sequences.

A292361 The number of paths of length 2m in the plane, starting and ending at (0,1), with unit steps in the four directions (north, east, south, west) and staying in the region y > 0 or x > -y.

1, 3, 21, 192, 2009, 22818, 273895, 3421318, 44042729, 580473551, 7796745921, 106365396629, 1470068855112, 20543335134692, 289818595800636, 4122517765350669, 59066177091706608

Offset: 0

Timothy Budd, Sep 15 2017

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

Cf. A135404.

a[n_] := SeriesCoefficient[-Pi(1 + 2 Sum[(y+3y^2+y^3)/(1+y+y^2+y^3+y^4) /. y->EllipticNomeQ[m]^l, {l,n+1}])/(4EllipticK[m]) /. m->16x, {x,0,n+1}]

G.f.: A(x) = 1/(2x) - (Pi / (4 x K(16x))) * (1 + 2 Sum_{n>=1} (q^n + 3q^(2n)+ q^(3n)) / (1 + q^n + q^(2n) + q^(3n) + q^(4n)) ), where q=q(16x) is the Jacobi nome of parameter m=16x and K(16x) is the complete elliptic integral of the first kind of parameter m=16x (proven).

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

Original entry on oeis.org

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

Offset: 2

Timothy Budd, Sep 14 2017

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.

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

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.

Cf. A005797.

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

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

cp's OEIS Frontend

User: Timothy Budd

A292361 The number of paths of length 2m in the plane, starting and ending at (0,1), with unit steps in the four directions (north, east, south, west) and staying in the region y > 0 or x > -y.

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