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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).

A272474 Triangle T(n,m) by rows: the number of tatami tilings of the 5 X n floor with dimers and m monomers.

Original entry on oeis.org

0, 3, 0, 4, 0, 1, 6, 0, 35, 0, 26, 0, 1, 0, 18, 0, 56, 0, 16, 3, 0, 52, 0, 64, 0, 7, 0, 10, 0, 88, 0, 80, 2, 0, 60, 0, 182, 0, 81, 0, 8, 0, 160, 0, 320, 0, 96, 2, 0, 102, 0, 500, 0, 449, 0, 112, 0, 18, 0, 340, 0, 952, 0, 600, 0, 120, 4, 0, 184, 0, 1056
Offset: 1

Views

Author

R. J. Mathar, Apr 30 2016

Keywords

Examples

			The triangle starts in row n=1 and column m=0 as:
0,3,0,4,0,1;
6,0,35,0,26,0,1;
0,18,0,56,0,16;
3,0,52,0,64,0,7;
0,10,0,88,0,80;
2,0,60,0,182,0,81;
0,8,0,160,0,320,0,96;
2,0,102,0,500,0,449,0,112;
0,18,0,340,0,952,0,600,0,120;
4,0,184,0,1056,0,1535,0,712,0,128;
0,24,0,550,0,2216,0,2338,0,824,0,128;
4,0,246,0,2050,0,4367,0,3256,0,936,0,128;
0,32,0,936,0,5044,0,7728,0,4454,0,1040,0,128;
6,0,414,0,4054,0,11539,0,12360,0,5816,0,1160,0,128;
0,52,0,1658,0,10736,0,22410,0,18744,0,7352,0,1280,0,128;
8,0,620,0,7412,0,27039,0,39590,0,26576,0,9056,0,1408,0,128;
0,68,0,2596,0,21180,0,57296,0,65634,0,36312,0,10864,0,1536,0,128;
10,0,908,0,13022,0,59625,0,112526,0,102054,0,47954,0,12816,0,1664,0,128;
0,100,0,4312,0,41056,0,138444,0,204496,0,152648,0,61720,0,14872,0,1792,0,128;
14,0,1404,0,23112,0,126291,0,298136,0,347122,0,219228,0,77904,0,17056,0,1920,0,128;
0,142,0,6904,0,77136,0,314464,0,584236,0,560856,0,305264,0,96552,0,19360,0,2048,0,128;
18,0,2034,0,38898,0,254427,0,731536,0,1068766,0,863460,0,413418,0,117944,0,21792,0,2176,0,128;
0,196,0,10778,0,139276,0,678728,0,1537620,0,1850598,0,1282412,0,546464,0,142128,0,24352,0,2304,0,128;
24,0,3018,0,65388,0,496213,0,1704232,0,3026128,0,3048168,0,1843736,0,707754,0,169288,0,27040,0,2432,0,128;

Links

  • R. J. Mathar, Re: Tatami, Seqfan mailing list of Mar 24 2016.

Crossrefs

Cf. A192091 (row sums), A068924 (column m=0), A281791 (column m=1), A272473 (4 by n grid).

Formula

G.f. x*( -3*x^11*y^8 +7*x^4*y^7 +x^2*y^7 -24*x^6*y^7 -2*x^5*y^4 -y^5 -2*x^3*y^2 -16*x^9 -9*x^ 2*y +3*x^3 +13*x^5 +7*x^15 -6*x -3*y -22*x*y^4 -32*x*y^2 +16*x^11*y^6 -6*x^13*y^2 -14*x^ 15*y^4 +3*x^13 +14*x^5*y^6 -11*x^11 +6*x^7 -55*y^3*x^10 -17*x^2*y^3 +37*x^6*y -24*y^3*x^14 -48*x^10*y +2*x^17*y^2 -19*x^12*y -4*y^3 -55*x^12*y^3 -8*x^11*y^4 +51*x^8*y^5 +28*x^7*y^6 +72*x^6*y^3 +31*y^2*x^15 -58*x^13*y^4 +x^17 -64*x^11*y^2 +36*x^10*y^5 +55*x^8*y^3 -50*x^6*y^5 -4*x^4*y^3 -5*x^9*y^4 -5*x^8*y +60*y^2*x^5 +84*x^7*y^2 -12*x^9*y^6 +55*x^7*y^4 -2*x^12*y^5 -79*x^9*y^2 -2*x^16*y^3 +4*x^7*y^8 +11*x^4*y -20*x^4*y^5 +6*x^13*y^ 6 +15*x^16*y +17*x^14*y^5 -5*x^10*y^7 -26*x^8*y^7 +x^8*y^9 +21*y*x^14 +11*x^2*y^5 +x^7* y^10 -x^10*y^9 +14*x^3*y^4 +9*x^3*y^6 -5*y^7*x^12 +3*x^9*y^8) / (x^14 +x^13*y +x^12 -2*x^ 12*y^2 -x^11*y^3 -2*x^10*y^2 -x^10 -x^9*y^3 +x^8*y^4 -3*x^8 -3*x^8*y^2 -x^7*y -x^7*y^3 +x^6*y^4 +4*x^6*y^2 -y^3*x^5 +2*x^4 +y^2*x^4 -x^3*y +x^2 +x*y -1). - R. J. Mathar, May 02 2016
G.f. for column m=1: x +14*x^3 +2*x*(1 +2*x^2 +3*x^4 -2*x^6 -4*x^8 -2*x^10)/ (1-x^4-x^6)^2. - R. J. Mathar, May 02 2016, corrected Apr 10 2017
G.f. for column m=2: -8 +17*x^2 +2*x^4 -2*(9*x^16 +24*x^14 +17*x^12 -22*x^10 -39*x^8 -9*x^6 +13*x^4 +9*x^2 +4) / (x^6+x^4-1)^3. - R. J. Mathar, May 02 2016

A192092 Number of tatami tilings of a 6 X n grid (with monomers allowed).

Original entry on oeis.org

1, 13, 156, 196, 238, 325, 450, 827, 1404, 2828, 4603, 7890, 12475, 20396, 34708, 57979, 102658, 170075, 292948, 482036, 812571, 1365010, 2293755, 3918292, 6555468, 11171195, 18648162, 31563547, 53005132, 89383740, 151102715
Offset: 0

Views

Author

Frank Ruskey and Yuji Yamauchi (eugene.uti(AT)gmail.com), Jul 14 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 6 X 3 grid:
    _ _ _ _ _ _   _ _ _ _ _ _   _ _ _ _ _ _   _ _ _ _ _ _
   |_ _| |_| |_| |_| |_|_ _| | | |_ _| |_| | |_| | |_ _| |
   |_ _|_| |_| | | |_|_ _| |_| |_| |_|_| |_| | |_|_| | |_|
   |_|_ _|_|_|_| |_|_ _|_|_|_| |_|_|_ _|_|_| |_|_ _|_|_|_|

Links

Crossrefs

Cf. A192091, A192093.
Showing 1-3 of 3 results.

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