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A271786 Expansion of 2*(1-x)*(2*x^2+4*x+1) / (1-x-x^2)^2.

%I A271786 #19 Dec 23 2024 14:53:44
%S A271786 2,10,18,38,72,136,250,454,814,1446,2548,4460,7762,13442,23178,39814,
%T A271786 68160,116336,198026,336254,569702,963270,1625708,2739028,4607522,
%U A271786 7739386,12982530,21750374,36396984,60839896,101593498,169482550,282481822,470419302
%N A271786 Expansion of 2*(1-x)*(2*x^2+4*x+1) / (1-x-x^2)^2.
%C A271786 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 A271786 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 A271786 R. J. Mathar, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2016-March/016246.html">Re: tatami</a>, SeqFan List of March 2016.
%H A271786 <a href="/index/Ta#tangent_numbers">Index entries related to Tatami mats</a>
%H A271786 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).
%F A271786 a(n) = 2*(A001629(n+2)+A271785(n)) .
%p A271786 A271786 := proc(n)
%p A271786     2*(A001629(n+2)+A271785(n)) ;
%p A271786 end proc:
%t A271786 LinearRecurrence[{2, 1, -2, -1}, {2, 10, 18, 38}, 34] (* _Jean-François Alcover_, Aug 08 2023 *)
%Y A271786 Cf. A001629, A271785, first column of A272472.
%K A271786 nonn,easy
%O A271786 0,1
%A A271786 _R. J. Mathar_, Apr 14 2016