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%I #18 Aug 08 2023 04:49:23
%S 2,10,18,38,72,136,250,454,814,1446,2548,4460,7762,13442,23178,39814,
%T 68160,116336,198026,336254,569702,963270,1625708,2739028,4607522,
%U 7739386,12982530,21750374,36396984,60839896,101593498,169482550,282481822,470419302
%N Expansion of 2*(1-x)*(2*x^2+4*x+1) / (1-x-x^2)^2.
%C The number of Tatami Tilings of the 3 X (2n+1) floor with one monomer at an arbitrary place (and therefore 3n+1 dimers).
%C The sequence is an overlay of the sequence b(n) = 1, 4, 7, 14, 26,... with g.f. B(x) = x*(1+2*x^2-2*x^4-2*x^6) / (1-x^2-x^4)^2 and the sequence c(n) = 0, 2, 4, 10, 20,... with g.f. C(x) = 2*x^3/(1-x^2-x^4)^2, meaning a(n) = 2*b(n)+c(n) = 2, 10, 18, 38, 72.... The sequence b(n) counts the tatami tilings with one monomer that must be in the first of the three lanes of the 3Xn grid. The sequence c(n) counts the tatami tilings with one monomer that must be in the middle lane of the grid. By up-down symmetry b(n) counts also the tatami tilings with one monomer that must be in the last of the three lanes. - _R. J. Mathar_, May 03 2016
%H R. J. Mathar, <a href="http://list.seqfan.eu/oldermail/seqfan/2016-March/016246.html">Re: tatami</a>, SeqFan List of March 2016.
%H <a href="/index/Ta#tangent_numbers">Index entries related to Tatami mats</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F a(n) = 2*(A001629(n+2)+A271785(n)) .
%p A271786 := proc(n)
%p 2*(A001629(n+2)+A271785(n)) ;
%p end proc:
%t LinearRecurrence[{2, 1, -2, -1}, {2, 10, 18, 38}, 34] (* _Jean-François Alcover_, Aug 08 2023 *)
%Y Cf. A001629, A271785, first column of A272472.
%K nonn,easy
%O 0,1
%A _R. J. Mathar_, Apr 14 2016
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