A266278 Number of legal Go positions on a 2 X n board.

5, 57, 489, 4125, 35117, 299681, 2557605, 21826045, 186255781, 1589441093, 13563736693, 115748216413, 987755062201, 8429158472781, 71931509371765, 613838505628281, 5238284505542721, 44701699729693429, 381468772192258129, 3255321946095461785, 27779786302899765081

Offset

1,1

Links

Colin Barker, Table of n, a(n) for n = 1..1000

John Tromp, Number of legal 2xn Go positions

J. Tromp and G. Farnebäck, Combinatorics of Go, Lecture Notes in Computer Science, 4630, 84-99, 2007.

Index entries for linear recurrences with constant coefficients, signature (10,-16,31,-13,20,2,-1).

Formula

a(n) = 10*a(n-1)-16*a(n-2)+31*a(n-3)-13*a(n-4)+20*a(n-5)+2*a(n-6)-a(n-7).

G.f.: x*(1 + x)^2*(5 - 3*x - 5*x^3 - x^4) / ((1 + x^2)*(1 - 10*x + 15*x^2 - 21*x^3 - 2*x^4 + x^5)). - Colin Barker, Jan 05 2018

Example

For n = 1, the a(1) = 5 legal 2 X 1 boards are .. X. O. .X .O

Prog

(PARI) Vec(x*(1 + x)^2*(5 - 3*x - 5*x^3 - x^4) / ((1 + x^2)*(1 - 10*x + 15*x^2 - 21*x^3 - 2*x^4 + x^5)) + O(x^40)) \\ Colin Barker, Jan 05 2018

Crossrefs

Cf. A094777, A102620.

Sequence in context: A196971 A197558 A218658 * A103047 A223628 A107339

Adjacent sequences: A266275 A266276 A266277 * A266279 A266280 A266281

Keyword

nonn,easy

Author

Felix Fröhlich, Dec 26 2015

Extensions

Corrected and edited by John Tromp, Jan 26 2016

Status

approved