A081113 Number of paths of length n-1 a king can take from one side of an n X n chessboard to the opposite side.

1, 4, 17, 68, 259, 950, 3387, 11814, 40503, 136946, 457795, 1515926, 4979777, 16246924, 52694573, 170028792, 546148863, 1747255194, 5569898331, 17698806798, 56076828573, 177208108824, 558658899825, 1757365514652

Offset

1,2

Comments

a(n) = number of sequences (a_1,a_2,...,a_n) with 1<=a_i<=n for all i and |a_(i+1)-a_(i)|<=1 for 1<=i<=n-1. For n=2 the sequences are 11, 12, 21, 22. - David Callan, Oct 24 2004

Simon Plouffe proposes the ordinary generating function A(x) (for offset zero) in the implicit form 3-10*x+12*x^2+(-4+30*x+54*x^3-72*x^2)*A(x)+(81*x^4+54*x^2+1-12*x-108*x^3)*A(x)^2 = 0 which delivers at least the first 200 terms (i.e., as far as tested) correctly. - David Scambler, R. J. Mathar, Jan 06 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Simon Plouffe, OEIS conjectured formulas.

D. Yaqubi, M. Farrokhi D. G., and H. Ghasemian Zoeram, Lattice paths inside a table, I , arXiv:1612.08697 [math.CO], 2016-2017.

Formula

a(n) = Sum_{k=1..n} k*(n-k+1)*M(n-1, k-1) where k*(n-k+1) is the triangular view of A003991 and M() is the Motzkin triangle A026300.

Conjecture: g.f.(x)=z*A064808(z), where z=x*A001006(x) and A...(x) are the corresponding generating functions. - R. J. Mathar, Jul 07 2009

Conjecture from WolframAlpha (verified for 1<=n<=180): (n+3)*a(n+4) = 27*n*a(n) +27*a(n+1) -9*(2*n+5)*a(n+2) +(8*n+21)*a(n+3). - Alexander R. Povolotsky, Jan 04 2011

Shorter recurrence: (n-1)*(2*n-7)*a(n) = (10*n^2-39*n+23)*a(n-1) - 3*(2*n^2-n-9)*a(n-2) - 9*(n-3)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 28 2012

a(n) ~ 3^(n-1)*n*(1-4/(sqrt(3*Pi*n))). - Vaclav Kotesovec, Oct 28 2012

a(n) = (n+2)*3^(n-2)+2*Sum_{k=0..n-3} (n-k-2)*3^(n-k-3)*A001006(k). [Yaqubi Corollary 2.8] - R. J. Mathar, Dec 13 2017

Example

For n=2 the 4 paths are (0,0)->(0,1); (0,0)->(1,1); (1,0)->(0,1); (1,0)->(1,1).

Maple

A026300 := proc(n, k) add( binomial(n, 2*i+n-k)*(binomial(2*i+n-k, i) -binomial(2*i+n-k, i-1)), i=0..floor(k/2)) ; end proc:
A081113 := proc(n) add(k*(n-k+1)*A026300(n-1, k-1), k=1..n) ; end proc:
seq(A081113(n), n=1..20) ;
# R. J. Mathar, Jun 09 2010

Mathematica

t[n_, k_] := Sum[ Binomial[n, 2i + n - k] (Binomial[2i + n - k, i] - Binomial[2i + n - k, i - 1]), {i, 0, Floor[k/2]}] (* from A026300 *); f[n_] := Sum[ k(n - k + 1)t[n - 1, k - 1], {k, n}]; Array[f, 24]

Crossrefs

Cf. A005773 (paths which begin at a corner), diagonal of A296449.

Sequence in context: A030529 A266862 A239845 * A368429 A114587 A268431

Adjacent sequences: A081110 A081111 A081112 * A081114 A081115 A081116

Keyword

easy,nonn

Author

David Scambler, Apr 16 2003

Status

approved