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

2, 10, 18, 38, 72, 136, 250, 454, 814, 1446, 2548, 4460, 7762, 13442, 23178, 39814, 68160, 116336, 198026, 336254, 569702, 963270, 1625708, 2739028, 4607522, 7739386, 12982530, 21750374, 36396984, 60839896, 101593498, 169482550, 282481822, 470419302

Offset: 0

Author

R. J. Mathar, Apr 14 2016

Keywords

• nonn
• easy

Comments

The number of Tatami Tilings of the 3 X (2n+1) floor with one monomer at an arbitrary place (and therefore 3n+1 dimers).

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

Links

• R. J. Mathar, Re: tatami, SeqFan List of March 2016.
• Index entries related to Tatami mats
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

Cf. A001629, A271785, first column of A272472.

Programs

Maple

A271786 := proc(n)
    2*(A001629(n+2)+A271785(n)) ;
end proc:

Mathematica

LinearRecurrence[{2, 1, -2, -1}, {2, 10, 18, 38}, 34] (* Jean-François Alcover, Aug 08 2023 *)

Formula

a(n) = 2*(A001629(n+2)+A271785(n)) .