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

0,1

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

Table of n, a(n) for n=0..33.

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).

Formula

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

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 *)

Crossrefs

Cf. A001629, A271785, first column of A272472.

Sequence in context: A173592 A018227 A092062 * A134251 A317714 A055260

Adjacent sequences: A271783 A271784 A271785 * A271787 A271788 A271789

Keyword

nonn,easy

Author

R. J. Mathar, Apr 14 2016

Status

approved