A318268 a(n) is the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that all but 3 such pairs are joined by an edge.

0, 0, 0, 2, 34, 250, 1234, 4830, 16174, 48444, 133416, 344220, 843020, 1978804, 4484228, 9865742, 21166390, 44439910, 91570126, 185614242, 370846914, 731502296, 1426514540, 2753525208, 5266164280, 9987859912, 18799814312, 35141997050, 65274659562, 120540177522

Offset

0,4

Comments

This is also the number of "(n-3)-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 23 2018

Links

Table of n, a(n) for n=0..29.

Andrew Howroyd, PARI program based on combinatorial definition

D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.

Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.

Index entries for linear recurrences with constant coefficients, signature (7,-17,11,19,-29,-3,21,-3,-7,1,1).

Formula

G.f.: x^2*(2*x + 20*x^2 + 46*x^3 + 40*x^4 + 30*x^5 + 4*x^6 + 4*x^7)/(1 - x)^3/(1 - x - x^2)^4 (conjectured).

The above conjecture is true. The PARI program given in the links can be used to establish an upper limit on the order of the linear recurrence and sufficient number of terms to prove correctness. - Andrew Howroyd, Sep 03 2018

Example

See example in A318267.

Mathematica

CoefficientList[Normal[Series[x^2(2*x + 20*x^2 + 46*x^3 + 40*x^4 + 30*x^5 + 4*x^6 + 4*x^7)/(1 - x)^3/(1 - x - x^2)^4, {x, 0, 30}]], x]
LinearRecurrence[{7, -17, 11, 19, -29, -3, 21, -3, -7, 1, 1}, {0, 0, 0, 2, 34, 250, 1234, 4830, 16174, 48444, 133416}, 30] (* Harvey P. Dale, Aug 05 2019 *)

Crossrefs

Cf. A046741, A318243, A318244, A318267, A318269, A318270.

Sequence in context: A064202 A206624 A131471 * A036827 A136362 A220507

Adjacent sequences: A318265 A318266 A318267 * A318269 A318270 A318271

Keyword

nonn,easy

Author

Donovan Young, Aug 22 2018

Extensions

Terms a(14) and beyond from Andrew Howroyd, Sep 03 2018

Status

approved