A096588 a(n)=number of Catalan knight paths in Quadrant I from (0,0) to points on the vertical line x=n. A Catalan knight moves (2 right and 1 up) or (1 right and 1 down).

1, 1, 3, 6, 16, 38, 99, 248, 646, 1659, 4342, 11307, 29740, 78115, 206349, 545156, 1445332, 3834559, 10197168, 27140709, 72357778, 193076677, 515843630, 1379308111, 3691755414, 9888374480, 26507373732, 71103941488, 190859621124

Offset

0,3

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022). Also arXiv:2206.12087, Jan 2023.

Formula

Row sums of array in A096587.

Example

Rows of array T(n,k) in A096587:
  1
  0 0 1
  1 1 0 0 1
  0 1 2 2 0 0 1
  ...
so a(3)=T(3,0)+T(3,1)+...+T(3,6)=6.

Mathematica

A096588list[nmax_]:=Module[{T}, T[0, 0]=1; T[n_, k_]:=T[n, k]=If[0<=k<=2n, T[n-1, k-2]+T[n-2, k-1]+T[n-1, k+2]+T[n-2, k+1], 0]; Table[Sum[T[n, k], {k, 0, 2n}], {n, 0, nmax}]]; A096588list[50] (* Paolo Xausa, May 22 2023 *)

Crossrefs

Cf. A096587.

Sequence in context: A321229 A114410 A190735 * A256943 A275207 A073079

Adjacent sequences: A096585 A096586 A096587 * A096589 A096590 A096591

Keyword

nonn

Author

Clark Kimberling, Jun 28 2004.

Status

approved