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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 5 such pairs are joined by an edge.
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%I #35 Mar 20 2020 15:09:11

%S 0,0,0,0,0,186,3666,36714,253386,1369260,6209700,24668742,88338174,

%T 290968686,894709790,2597386330,7181246394,19040425628,48684375292,

%U 120592523460,290476059204,682548818802,1568744083242,3534725236308,7823387477220,17037467831748

%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 5 such pairs are joined by an edge.

%C This is also the number of "(n-5)-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - _Donovan Young_, Oct 23 2018

%H D. Young, <a href="http://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_17">Index entries for linear recurrences with constant coefficients</a>, signature (11,-49,105,-75,-123,278,-82,-250,210,90,-150,-5,55,-5,-11,1,1).

%F G.f.: x^2*(6*x^13 + 20*x^12 + 228*x^11 + 888*x^10 + 3012*x^9 + 6612*x^8 + 10020*x^7 + 9636*x^6 + 5502*x^5 + 1620*x^4 + 186*x^3)/(1 - x)^5/(1 - x - x^2)^6 (conjectured).

%F The above conjecture is true. See A318268. - _Andrew Howroyd_, Sep 03 2018

%e See example in A318267.

%t CoefficientList[Normal[Series[x^2(6*x^13+20*x^12+228*x^11+888*x^10+3012*x^9+6612*x^8+10020*x^7+9636*x^6+5502*x^5+1620*x^4+186*x^3)/(1-x)^5/(1-x-x^2)^6,{x,0,30}]],x]

%Y Cf. A046741, A318243, A318244, A318267, A318268, A318269.

%K nonn,easy

%O 0,6

%A _Donovan Young_, Aug 23 2018

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