A025609 -id:A025609

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A025607

Number of n-move rook paths on 8 X 8 board from given corner to same corner.

+10
3

1, 0, 14, 84, 896, 10080, 127904, 1708224, 23426816, 325032960, 4532831744, 63353816064, 886318555136, 12404650352640, 173642248822784, 2430854346031104, 34031138021113856, 476430995352453120, 6670004313281921024, 93379882656019513344, 1307317290804734590976

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Paths are not required to be self-avoiding. - Andrew Howroyd, Nov 05 2019

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (18,-44,-168).

FORMULA

G.f.: (1 - 18*x + 58*x^2)/((1 + 2*x)*(1 - 6*x)*(1 - 14*x)).

a(n) = 18*a(n-1) - 44*a(n-2) - 168*a(n-3) for n >= 3. - Andrew Howroyd, Nov 05 2019

MATHEMATICA

CoefficientList[ Series[(1 - 18 x + 58 x^2)/((1 + 2 x) (1 - 6 x) (1 - 14 x)), {x, 0, 16}], x]

LinearRecurrence[{18, -44, -168}, {1, 0, 14}, 30] (* Harvey P. Dale, Feb 13 2023 *)

PROG

(PARI) Vec((1 - 18*x + 58*x^2)/((1 + 2*x)*(1 - 6*x)*(1 - 14*x)) + O(x^20)) \\ Andrew Howroyd, Nov 05 2019

CROSSREFS

Cf. A025608, A025609.

KEYWORD

nonn,easy,walk

AUTHOR

David W. Wilson

EXTENSIONS

Terms a(17) and beyond from Andrew Howroyd, Nov 05 2019

STATUS

approved

A025608

Number of n-move rook paths on 8 X 8 board from given corner to opposite corner.

+10
3

0, 0, 2, 36, 560, 8160, 116192, 1638336, 23006720, 322513920, 4517714432, 63263118336, 885774356480, 12401385185280, 173622657769472, 2430736799809536, 34030432743587840, 476426763687690240, 6669978923292557312, 93379730316084903936, 1307316376765123788800

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Paths are not required to be self-avoiding. - Andrew Howroyd, Nov 05 2019

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (18,-44,-168).

FORMULA

G.f.: (2*x^2)/((1 + 2*x)*(1 - 6*x)*(1 - 14*x)).

a(0)=0, a(1)=0, a(2)=2, a(n)=18*a(n-1)-44*a(n-2)-168*a(n-3). - Harvey P. Dale, Mar 09 2013

MATHEMATICA

CoefficientList[Series[(2x^2)/((1+2x)(1-6x)(1-14x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{18, -44, -168}, {0, 0, 2}, 30] (* Harvey P. Dale, Mar 09 2013 *)

PROG

(PARI) concat([0, 0], Vec(2/((1 + 2*x)*(1 - 6*x)*(1 - 14*x)) + O(x^20))) \\ Andrew Howroyd, Nov 05 2019

CROSSREFS

Cf. A025607, A025609.