A026327 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 4. Also a(n) = T(n,n-2), where T is the array in A026323.

1, 3, 10, 30, 90, 266, 783, 2295, 6710, 19580, 57057, 166101, 483210, 1405080, 4084590, 11872494, 34508997, 100313635, 291646580, 848102640, 2466916474, 7177785582, 20891443950, 60827142350, 177167486925, 516217883571, 1504692189588

Offset

2,2

Comments

Number of paths in the plane x>=0 and y>=-2, from (0,0) to (n,2), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=4, we have the 10 paths: UUUD, UUHH, UUDU, UHUH, UHHU, UDUU, HUUH, HUHU, HHUU, DUUU. - José Luis Ramírez Ramírez, Apr 20 2015

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000

Formula

Conjecture: -(n+8)*(n-2)*a(n) +3*(2*n^2+7*n-28)*a(n-1) +3*(-3*n^2-n+36)*a(n-2) -4*(n+4)*(n-1)*a(n-3) +12*(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

G.f: x^2*M(x)^2/(1-x-x^2*(M(x)+1/(1-x-x^2/(1-x)))), where M(x) is g.f. of Motzkin paths. - José Luis Ramírez Ramírez, Apr 20 2015

a(n) ~ 5 * 3^(n+5/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

Mathematica

Rest[Rest[CoefficientList[Series[x^2*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2))^2/(1-x-x^2*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)+1/(1-x-x^2/(1-x)))), {x, 0, 20}], x]]] (* Vaclav Kotesovec, Apr 21 2015 *)

Crossrefs

Sequence in context: A257596 A261336 A026109 * A014531 A062107 A269800

Adjacent sequences: A026324 A026325 A026326 * A026328 A026329 A026330

Keyword

nonn

Author

Clark Kimberling

Status

approved