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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.
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%I #43 Mar 20 2020 15:08:56

%S 0,0,0,2,34,250,1234,4830,16174,48444,133416,344220,843020,1978804,

%T 4484228,9865742,21166390,44439910,91570126,185614242,370846914,

%U 731502296,1426514540,2753525208,5266164280,9987859912,18799814312,35141997050,65274659562,120540177522

%N 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.

%C 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

%H Andrew Howroyd, <a href="/A318268/a318268.txt">PARI program based on combinatorial definition</a>

%H D. Young, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Young/young2.html">The Number of Domino Matchings in the Game of Memory</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.

%H Donovan Young, <a href="https://arxiv.org/abs/1905.13165">Generating Functions for Domino Matchings in the 2 * k Game of Memory</a>, arXiv:1905.13165 [math.CO], 2019. Also in <a href="https://www.emis.de/journals/JIS/VOL22/Young/young13.html">J. Int. Seq.</a>, Vol. 22 (2019), Article 19.8.7.

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (7,-17,11,19,-29,-3,21,-3,-7,1,1).

%F 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).

%F 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

%e See example in A318267.

%t 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]

%t 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 *)

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

%K nonn,easy

%O 0,4

%A _Donovan Young_, Aug 22 2018

%E Terms a(14) and beyond from _Andrew Howroyd_, Sep 03 2018