A068924 -id:A068924 - cp's OEIS frontend

A068920 Table of t(r,s) read by antidiagonals: t(r,s) is the number of ways to tile an r X s room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 4, 0, 4, 0, 1, 6, 4, 4, 6, 1, 0, 9, 0, 2, 0, 9, 0, 1, 13, 6, 3, 3, 6, 13, 1, 0, 19, 0, 3, 0, 3, 0, 19, 0, 1, 28, 10, 3, 2, 2, 3, 10, 28, 1, 0, 41, 0, 5, 0, 2, 0, 5, 0, 41, 0, 1, 60, 16, 5, 2, 2, 2, 2, 5, 16, 60, 1, 0, 88, 0, 6, 0, 1, 0, 1, 0, 6, 0, 88, 0, 1, 129, 26

Offset: 1

Dean Hickerson, Mar 11 2002

Rows 2-6 are given in A068921 - A068925.

			Table begins:
  0, 1, 0, 1, 0, 1, ...
  1, 2, 3, 4, 6, 9, ...
  0, 3, 0, 4, 0, 6, ...
  1, 4, 4, 2, 3, 3, ...
  0, 6, 0, 3, 0, 2, ...
  1, 9, 6, 3, 2, 2, ...
  ...

Dean Hickerson, Filling rectangular rooms with Tatami mats

Cf. A068926 for incongruent tilings, A067925 for count by area.

Cf. A068921 (row 2), A068922 (row 3), A068923 (row 4), A068924 (row 5), A068925 (row 6).

(* See link for Mathematica programs. *)
c[r_, s_] := Which[s<0, 0, r==1, 1 - Mod[s, 2], r == 2, c1[2, s] + c2[2, s] + Boole[s == 0], OddQ[r], c[r, s] = c[r, s - r + 1] + c[r, s - r - 1] + Boole[s == 0], EvenQ[r], c[r, s] = c1[r, s] + c2[r, s] + Boole[s == 0]];
c1[r_, s_] := Which[s <= 0, 0, r == 2, c[2, s - 1], EvenQ[r], c2[r, s - 1] + Boole[s == 1]];
c2[r_, s_] := Which[s <= 0, 0, r == 2, c2[2, s] = c1[2, s - 2] + Boole[s == 2], EvenQ[r], c2[r, s] = c1[r, s - r + 2] + c1[r, s - r] + Boole[s == r - 2] + Boole[s == r]];
t[r_, s_] := Which[r>s, t[s, r], OddQ[r] && r>1, 2 c[r, s], True, c[r, s]];
A068920[n_] := Module[{x}, x = Floor[(Sqrt[8 n + 1] - 1)/2]; t[n + 1 - x (x + 1)/2, (x + 1) (x + 2)/2 - n]];
Table[A068920[n], {n, 0, 100}] (* Jean-François Alcover, May 12 2017, copied and adapted from Dean Hickerson's programs *)

A068930 Number of incongruent ways to tile a 5 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Original entry on oeis.org

4, 2, 1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 9, 13, 15, 22, 26, 37, 45, 63, 78, 108, 136, 186, 237, 322, 414, 559, 724, 973, 1267, 1697, 2219, 2964, 3888, 5183, 6815, 9071, 11949, 15886, 20955, 27835, 36755, 48790, 64476, 85545, 113115, 150021, 198460, 263136

Offset: 1

Dean Hickerson, Mar 11 2002

R. J. Mathar, Paving rectangular regions with rectangular tiles,...., arXiv:1311.6135 [math.CO], Table 12.
Index entries for linear recurrences with constant coefficients, signature (0, 1, 1, 1, 0, 0, -1, -1, -1).

Cf. A068924 for total number of tilings, A068926 for more info.

Cf. A005683.

Join[{4,2},LinearRecurrence[{0,1,1,1,0,0,-1,-1,-1},{1,1,1,2,2,3,3,5,5},50]] (* Harvey P. Dale, Nov 21 2014 *)

For n >= 12, a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-7) - a(n-8) - a(n-9).
G.f.: x*(4+x^10+5*x^9+4*x^8+3*x^7-x^6-2*x^5-6*x^4-5*x^3 -3*x^2+2*x) / ((x^3+x^2-1)*(x^6+x^4-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009
a(n) = sum(A102541(n-k-2, n-2*k-4), k=0..floor((n-4)/2)), n >= 4. - Johannes W. Meijer, Aug 24 2013

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

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

R. J. Mathar, Apr 30 2016

			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;

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

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

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

A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)(x^3+2x^2+x-1) ).

Original entry on oeis.org

1, 1, 1, 4, 9, 16, 36, 81, 169, 361, 784, 1681, 3600, 7744, 16641, 35721, 76729, 164836, 354025, 760384, 1633284, 3508129, 7535025, 16184529, 34762816, 74666881, 160376896, 344473600, 739894401, 1589218225, 3413480625, 7331811876, 15747991081, 33825095056

Offset: 0

R. J. Mathar, Dec 06 2013

a(n) is the number of tilings of a 3 X 2 X n room with bricks of 1 X 1 X 3 shape (and in that respect a generalization of A028447 which fills 3 X 2 X n rooms with bricks of 1 X 1 X 2 shape).
The inverse INVERT transform is 1, 0, 3, 2, 2, 4, 4, 6, 8, 10, .. , continued as in A068924.
a(n) is the number of tilings of an n-board (a board with dimensions n X 1) with half-squares (1/2 X 1 pieces, always placed so that the shorter sides are horizontal) and (1/2,1/2;3)-combs. A (w,g;m)-comb is a tile composed of m pieces of dimensions w X 1 separated horizontally by gaps of width g. - Michael A. Allen, Sep 24 2024

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Connections between two classes of generalized Fibonacci numbers squared and permanents of (0,1) Toeplitz matrices, Lin. Multilin. Alg. 72:13 (2024) 2091-2103.
R. J. Mathar, Tilings of rectangular regions by rectangular tiles: counts derived from transfer matrices, arXiv:1406.7788 [math.CO], 2014, see eq. (39).
Index entries for linear recurrences with constant coefficients, signature (1,1,3,1,-1,-1).

Cf. A000930.

A233247 := proc(n)
    A000930(n)^2 ;
end proc:
# second Maple program:
a:= n-> (<<0|1|0>, <0|0|1>, <1|0|1>>^n)[3, 3]^2:
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 06 2013

Table[Sum[Binomial[n-2i, i], {i,0,n/3}]^2, {n,0,50}] (* Wesley Ivan Hurt, Dec 06 2013 *)
LinearRecurrence[{1,1,3,1,-1,-1},{1,1,1,4,9,16},40] (* Harvey P. Dale, Jan 14 2015 *)
CoefficientList[Series[(1-x^3-x^2)/((x^3-x^2-1)*(x^3+2*x^2+x-1)), {x, 0, 50}], x] (* G. C. Greubel, Apr 29 2017 *)

my(x='x+O('x^50)); Vec((1-x^3-x^2)/((x^3-x^2-1)*(x^3+2*x^2+x-1))) \\ G. C. Greubel, Apr 29 2017

a(n) = A000930(n)^2.

a(n) = a(n-1) + a(n-3) + 2*Sum_{r=3..n} ( A000931(r+2)*a(n-r) ). - Michael A. Allen, Sep 24 2024

A232621 The number of vertically fault-free domino tilings of the 5 X (2n) board.

Original entry on oeis.org

1, 8, 31, 175, 1015, 5911, 34447, 200767, 1170151, 6820135, 39750655, 231683791, 1350352087, 7870428727, 45872220271, 267362892895, 1558305137095, 9082467929671, 52936502440927, 308536546715887, 1798282777854391, 10481160120410455, 61088677944608335

Offset: 0

R. J. Mathar, Nov 27 2013

A003775 counts the tilings of the 5 X (2n) board, and this sequence here counts only those that cannot be broken into tilings of two or more smaller 5 X (2n') boards with edge lengths n' < n by cutting "vertically" through the tiling parallel to the "short" side of length 5.

Technically speaking this is the inverse INVERT transform of A003775 (see the comment in A005178).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A003775, A068924.

Vec((-18*x^2+13*x^3-x^4+x+1)/((1-x)*(1-6*x+x^2)) + O(x^30)) \\ Colin Barker, Mar 05 2016

G.f.: (1 + x - 18*x^2 + 13*x^3 - x^4)/((1-x)*(1 - 6*x + x^2)).
a(n) = 1 + 6*A001653(n) for n>1. - Bruno Berselli, Nov 27 2013
a(n) = 6*a(n-1) - a(n-2) - 4, n>=4. - R. J. Mathar, Nov 07 2015
a(n) = 1 + (3/2)*(3-2*sqrt(2))^n*(2+sqrt(2)) + (3-3/sqrt(2))*(3+2*sqrt(2))^n for n>1. - Colin Barker, Mar 05 2016

cp's OEIS Frontend

A068920 Table of t(r,s) read by antidiagonals: t(r,s) is the number of ways to tile an r X s room with 1 X 2 Tatami mats. At most 3 Tatami mats may meet at a point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A068930 Number of incongruent ways to tile a 5 X 2n room with 1x2 Tatami mats. At most 3 Tatami mats may meet at a point.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)*(x^3+2*x^2+x-1) ).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A232621 The number of vertically fault-free domino tilings of the 5 X (2n) board.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A233247 Expansion of ( 1-x^3-x^2 ) / ( (x^3-x^2-1)(x^3+2x^2+x-1) ).