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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A192090 Number of tatami tilings of a 4 X n grid (with monomers allowed).

Original entry on oeis.org

1, 5, 29, 44, 66, 126, 238, 490, 922, 1714, 3306, 6246, 12102, 22994, 43682, 83810, 159154, 305062, 581382, 1108362, 2119602, 4037338, 7716554, 14720142, 28084702, 53639778, 102298794, 195341594, 372753634, 711338798, 1357975774
Offset: 0

Views

Author

Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jun 23 2011

Keywords

Comments

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

Examples

			Here are some tatami tilings of the 4 X 3 grid:
    _ _ _ _   _ _ _ _   _ _ _ _   _ _ _ _
   |_ _| |_| |_| |_ _| | |_ _| | |_| |_ _|
   |_ _|_| | | |_|_ _| |_| |_|_| | |_|_ _|
   |_|_ _|_| |_|_ _|_| |_|_|_ _| |_|_ _|_|

Links

Crossrefs

Cf. A180970, (3 X n grid), A192091 (5 X n grid), row sums of A272473.

Formula

G.f.: -13 + 3*x + 3*x^2 + 2*x^3 + (14 - 12*x + 10*x^2 + 10*x^4 - 104*x^5 + 114*x^6 - 80*x^7 + 34*x^8 + 12*x^9 - 2*x^10)/(1 - x - x^2 - x^3 + x^4 - 7*x^5 + 7*x^6 - x^7 + x^8 + x^9 + x^10 - x^11).

A272472 Triangle T(n,m) by rows: The number of tatami tilings of a 3 by n grid with dimers and m monomers.

Original entry on oeis.org

0, 2, 0, 1, 3, 0, 9, 0, 1, 0, 10, 0, 12, 4, 0, 27, 0, 13, 0, 18, 0, 56, 0, 16, 6, 0, 75, 0, 97, 0, 18, 0, 38, 0, 198, 0, 152, 0, 18, 10, 0, 177, 0, 433, 0, 214, 0, 18, 0, 72, 0, 570, 0, 836, 0, 282, 0, 18, 16, 0, 393, 0, 1517, 0, 1442, 0, 354, 0, 18, 0, 136
Offset: 1

Views

Author

R. J. Mathar, Apr 30 2016

Keywords

Examples

			The triangle starts in row n=1 and column m=0 as:
0,2,0,1;
3,0,9,0,1;
0,10,0,12;
4,0,27,0,13;
0,18,0,56,0,16;
6,0,75,0,97,0,18;
0,38,0,198,0,152,0,18;
10,0,177,0,433,0,214,0,18;
0,72,0,570,0,836,0,282,0,18;
16,0,393,0,1517,0,1442,0,354,0,18;
0,136,0,1478,0,3472,0,2292,0,426,0,18;
26,0,829,0,4571,0,7052,0,3410,0,498,0,18;
0,250,0,3554,0,12070,0,13076,0,4808,0,570,0,18;
42,0,1691,0,12479,0,28158,0,22480,0,6494,0,642,0,18;
0,454,0,8108,0,37222,0,59530,0,36308,0,8468,0,714,0,18;
68,0,3359,0,31729,0,97766,0,115948,0,55672,0,10730,0,786,0,18;
0,814,0,17768,0,105238,0,231622,0,210880,0,81708,0,13280,0,858,0,18;
110,0,6537,0,76483,0,306606,0,503348,0,361878,0,115568,0,16118,0,930,0,18;
0,1446,0,37736,0,278626,0,803060,0,1016880,0,590846,0,158404,0,19244,0,1002,0,18;
178,0,12511,0,176833,0,889916,0,1923278,0,1929730,0,924216,0,211368,0,22658,0,1074,0,18;
0,2548,0,78144,0,700670,0,2549216,0,4268026,0,3469042,0,1392996,0,275612,0,26360,0,1146,0,18;
288,0,23617,0,395387,0,2430464,0,6661414,0,8867630,0,5948792,0,2032802,0,352288,0,30350,0,1218,0,18;
0,4460,0,158492,0,1690478,0,7547920,0,16089358,0,17395888,0,9787628,0,2883858,0,442548,0,34628,0,1290,0,18;
466,0,44067,0,860069,0,6319840,0,21344172,0,36292416,0,32446518,0,15527142,0,3990996,0,547544,0,39194,0,1362,0,18;

Links

Crossrefs

Cf. A180970 (row sums), A068922 (column m=0), A271786 (column m=1), A272471 (2 by n grid), A100245 (row reversed without tatami condition).

Formula

G.f. x *(x^4*y^3 +7*x*y^2 +3*x +2*y +y^3 +x^6*y +3*x^2*y -x^3*y^2 -6*x^4*y -x^2*y^5 +x^2*y^3 +y^3*x^6 -2*y^4*x^5 -2*x^3 -2*x^5 +y^5*x^4 -x^3*y^4 -x^5*y^2 +x^7) / (x^6 +x^5*y -2*x^4*y^2 -2*x^2 -x*y +1). - R. J. Mathar, May 01 2016

A180965 Number of tatami tilings of a 2 X n grid (with monomers allowed).

Original entry on oeis.org

1, 2, 6, 13, 29, 68, 156, 357, 821, 1886, 4330, 9945, 22841, 52456, 120472, 276681, 635433, 1459354, 3351598, 7697381, 17678037, 40599916, 93242996, 214144685, 491811165, 1129508406, 2594063186, 5957604017, 13682413681, 31423445328, 72168035504, 165743294353
Offset: 0

Views

Author

Frank Ruskey, Sep 29 2010

Keywords

Comments

A tatami tiling consists of dimers (1 x 2) and monomers (1 x 1) where no four meet at a point.
Also, a(n) is the number of permutations of 1, ..., n+1 such that i can go to j only if |i-j| <= 2 and such that the pattern cdab (two consecutive pairs of elements swap position) is explicitly forbidden. - Jean M. Morales, Jun 02 2013

Examples

			Below we show the a(3) = 13 tatami tilings of a 2 X 3 rectangle where v = square of a vertical dimer, h = square of a horizontal dimer, m = monomer:
vvv vhh vmm vhh vmv vvm hhv hhm hhv mvv mhh mmv mvm
vvv vhh vhh vmm vmv vvm hhv mhh mmv mvv hhm hhv mvm

Links

Crossrefs

Cf. A000045 (1 X n grid), A180970 (3 X n grid).
Row sums of A272471.

Programs

  • GAP
    a:=[1,2,6,13];; for n in [5..35] do a[n]:=2*a[n-1]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 11 2018
  • Magma
    I:=[1,2,6,13]; [n le 4 select I[n] else 2*Self(n-1)+2*Self(n-3)-Self(n-4): n in [1..35]]; // Vincenzo Librandi, Sep 11 2018
  • Maple
    seq(coeff(series((1+2*x^2-x^3)/(x^4-2*x^3-2*x+1),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Sep 11 2018
  • Mathematica
    CoefficientList[(1+2z^2-z^3)/(1-2z-2z^3+z^4) + O[z]^32, z] (* Jean-François Alcover, May 27 2015 *)
LinearRecurrence[{2,0,2,-1},{1,2,6,13},40] (* Harvey P. Dale, Jan 19 2023 *)
  • PARI
    my(x='x+O('x^50)); Vec((1+2*x^2-x^3)/(1-2*x-2*x^3+x^4)) \\ Altug Alkan, Sep 10 2018
  • Python
    from math import log
print(0,1)
print(1,2)
print(2,6)
print(3,13)
n,a0,a1,a2,a3 = 3,13,6,2,1
while log(a0)/log(10) < 1000:
    n,a0,a1,a2,a3 = n+1,2*(a0+a2)-a3,a0,a1,a2
    print(n,a0) # A.H.M. Smeets, Sep 10 2018
  • Sage
    def A180965_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+2*x^2-x^3)/(1-2*x-2*x^3+x^4) ).list()
A180965_list(40) # G. C. Greubel, Apr 06 2021

Formula

G.f.: (1 + 2*x^2 - x^3)/(1 - 2*x - 2*x^3 + x^4).
Lim_{n -> inf} a(n)/a(n-1) = (sqrt(3) + 1)/2 + sqrt(sqrt(3)/2). - A.H.M. Smeets, Sep 10 2018
a(n) = 2*a(n-1) + 2*a(n-3) - a(n-4). - Muniru A Asiru, Sep 11 2018
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