A272472 -id:A272472 - cp's OEIS frontend

A068922 Number of ways to tile a 3 X 2n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

3, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338, 126491972

Offset: 1

Dean Hickerson, Mar 11 2002

Colin Barker, Table of n, a(n) for n = 1..1000
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 2.
F. Ruskey and J. Woodcock, Counting Fixed-Height Tatami Tilings, Electronic Journal of Combinatorics, Paper R126 (2009) 20 pages.
Index entries for linear recurrences with constant coefficients, signature (1,1).

Cf. A068928 for incongruent tilings, A068920 for more info. First column of A272472.
Essentially the same as A006355.
Essentially the same as A078642. - Georg Fischer, Oct 06 2018

Concatenation([3],List([2..40],n->2*Fibonacci(n+1))); # Muniru A Asiru, Oct 07 2018

[3] cat [2*Fibonacci(n+1): n in [2..50]]; // Vincenzo Librandi, Oct 07 2018

with(combinat): 3,seq(2*fibonacci(n+1),n=2..40); # Muniru A Asiru, Oct 07 2018

Join[{3}, Table[2 Fibonacci[n + 1], {n, 2, 50}]] (* Vincenzo Librandi, Oct 07 2018 *)
CoefficientList[Series[(x^2-x-3) / (x^2+x-1), {x, 0, 50}], x] (* Stefano Spezia, Oct 07 2018 *)

Vec(x*(3+x-x^2) / (1-x-x^2) + O(x^50)) \\ Colin Barker, Jan 29 2017

For n >= 2, a(n) = 2*F(n+1), where F(n)=A000045(n) is the n-th Fibonacci number.
G.f.: x*(x^2-x-3) / (x^2+x-1). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; checked and corrected by R. J. Mathar, Sep 16 2009
From Colin Barker, Jan 29 2017: (Start)
a(n) = (2^(-n)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / sqrt(5) for n>1.
a(n) = a(n-1) + a(n-2) for n>3. (End)
E.g.f.: 2*exp(x/2)*(5*cosh(sqrt(5)*x/2) + sqrt(5)*sinh(sqrt(5)*x/2))/5 - 2 + x. - Stefano Spezia, Apr 18 2022

A180970 Number of tatami tilings of a 3 X n grid (with monomers allowed).

Original entry on oeis.org

1, 3, 13, 22, 44, 90, 196, 406, 852, 1778, 3740, 7822, 16404, 34346, 72004, 150822, 316076, 662186, 1387596, 2907262, 6091780, 12763778, 26744268, 56036566, 117413804, 246015450, 515476036, 1080072022, 2263070868, 4741795442

Offset: 0

Frank Ruskey, Sep 29 2010

nonn

A tatami tiling consists of dimers (1 X 2) and monomers (1 X 1) where no four meet at a point.

			Below we show the a(2) = 13 tatami tilings of a 2 X 3 rectangle where v = square of a vertical dimer, h = square of a horizontal dimer, m = monomer:
hh hh hh hh hh hh vv vm vm mm mv mv mm
hh vv mv vm mm hh vv vv vm hh vv mv hh
hh vv mv vm hh mm hh mv hh hh vm hh mm

A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, Auspicious Tatami Mat Arrangements, The 16th Annual International Computing and Combinatorics Conference (COCOON 2010), July 19-21, Nha Trang, Vietnam. LNCS 6196 (2010) 288-297.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Erickson, F. Ruskey, M. Schurch and J. Woodcock, Monomer-Dimer Tatami Tilings of Rectangular Regions, Electronic Journal of Combinatorics, 18(1) (2011) P109.
Alejandro Erickson, Frank Ruskey, Mark Schurch, and Jennifer Woodcock, Auspicious tatami mat arrangements, arXiv:1103.3309 [math.CO], 2011. See p. 17.
Index entries for linear recurrences with constant coefficients, signature (1,2,0,2,-1,-1).

Cf. A180965 (2 X n grid), A192090 (4 X n grid), row sums of A272472.

R:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1 +2*x +8*x^2 +3*x^3 -6*x^4 -3*x^5 -4*x^6 +2*x^7 +x^8)/(1 -x -2*x^2 -2*x^4 +x^5 +x^6) )); // G. C. Greubel, Apr 05 2021

Join[{1,3,13}, LinearRecurrence[{1,2,0,2,-1,-1}, {22,44,90,196,406,852}, 37]] (* Jean-François Alcover, Jan 29 2019 *)

def A180970_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1 +2*x +8*x^2 +3*x^3 -6*x^4 -3*x^5 -4*x^6 +2*x^7 +x^8)/(1 -x -2*x^2 -2*x^4 +x^5 +x^6) ).list()
A180970_list(40) # G. C. Greubel, Apr 05 2021

G.f.: (1 + 2*x + 8*x^2 + 3*x^3 - 6*x^4 - 3*x^5 - 4*x^6 + 2*x^7 + x^8)/(1 - x - 2*x^2 - 2*x^4 + x^5 + x^6).

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

Original entry on oeis.org

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

R. J. Mathar, Apr 14 2016

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

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).

Cf. A001629, A271785, first column of A272472.

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

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

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

A272473 Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

Original entry on oeis.org

1, 3, 1, 4, 18, 7, 4, 27, 13, 2, 32, 32, 3, 52, 64, 7, 3, 62, 133, 40, 3, 99, 269, 110, 9, 5, 152, 437, 280, 48, 5, 163, 730, 669, 138, 9, 6, 258, 1243, 1318, 433, 48, 8, 343, 1823, 2670, 1239, 154, 9, 8, 408, 2949, 5240, 2849, 600, 48, 11, 632, 4577

Offset: 1

R. J. Mathar, Apr 30 2016

The number of squares in the 4 by n floor is even, so the number of tilings with an odd number of monomers is zero.

			The triangle starts in row n=1 and column m=0 as:
1,3,1;
4,18,7;
4,27,13;
2,32,32;
3,52,64,7;
3,62,133,40;
3,99,269,110,9;
5,152,437,280,48;
5,163,730,669,138,9;
6,258,1243,1318,433,48;
8,343,1823,2670,1239,154,9;
8,408,2949,5240,2849,600,48;
11,632,4577,9011,6655,1927,172,9;
13,746,6287,16184,14697,4930,777,48;
14,971,9928,28135,28805,13089,2669,190,9;
19,1394,14234,44806,58022,32176,7501,954,48;
21,1610,19501,75702,111795,70427,22344,3445,208,9;
25,2224,29785,121302,199354,157078,59859,10576,1131,48;
32,2909,40073,184597,366553,331449,143611,34646,4257,226,9;
35,3464,55939,298278,644436,651772,350855,99300,14167,1308,48;
44,4820,81474,449995,1081033,1303651,802565,258303,50095,5105,244,9;
53,5924,106460,670726,1868914,2488996,1719501,684338,151835,18274,1485,48;
60,7408,150672,1040424,3077401,4548409,3716945,1678785,425017,68761,5989,262,9;
76,9972,208211,1503372,4956628,8434302,7641320,3879356,1208052,218806,22897,1662,48;

Cf. A192090 (row sums), A068923 (column m=0), A272472 (3 by n grid), A100265 (without tatami condition, reversed rows).

G.f. x*( -1 -8*x^7*y^2 +21*x^5*y^2 -7*x^7*y^6 +4*x^3*y^2 -3*x^7 +2*x^5 -8*x^2*y^2 -4*x^8*y^4 -3*x -6*x*y^4 -15*x*y^2 -2*x^3*y^4 -6*x^8 -5*x^10*y^2 -5*x^9*y^2 -y^4 -2*x^8*y^2 -3*y^2 -8*x^11*y^2 +5*x^11*y^4 -3*x^2*y^4 -2*x^5*y^6 +2*x^13 +x^12 +x^11 +x^6 -7*x^7*y^4 +x^7*y^8 +11*x^4*y^2 -3*x^9 -15*x^10*y^4 -2*x^10*y^6 +18*x^9*y^4 +36*x^6*y^4 +20*x^6*y^2 -17*x^ 5*y^4 -8*x^4*y^4 +4*x^3 +8*x^6*y^6 +5*x^4 +2*x^9*y^6 -y^8*x^6 +6*y^6*x^3 +y^6*x^2)/ (x^11 -x^10 +2*x^9 -3*x^9*y^2 +x^8*y^2 -2*x^8 +x^7 +x^6*y^4 -5*x^6*y^2 -3*x^6 +2*x^5 +5*x^5*y^2 +x^4*y^2 -2*x^4 -x^3*y^2 +2*x^3 +x^2*y^2 +x -1). - R. J. Mathar, May 01 2016

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A068922 Number of ways to tile a 3 X 2n room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

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A180970 Number of tatami tilings of a 3 X n grid (with monomers allowed).

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

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A272473 Triangle T(n,m) by rows: the number of tatami tilings of a 4 by n grid with 2*m monomers.

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Formula

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