A028468 -id:A028468 - cp's OEIS frontend

A099390 Array T(m,n) read by antidiagonals: number of domino tilings (or dimer tilings) of the m X n grid (or m X n rectangle), for m>=1, n>=1.

0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 5, 0, 5, 0, 1, 8, 11, 11, 8, 1, 0, 13, 0, 36, 0, 13, 0, 1, 21, 41, 95, 95, 41, 21, 1, 0, 34, 0, 281, 0, 281, 0, 34, 0, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 0, 89, 0, 2245, 0, 6728, 0, 2245, 0, 89, 0, 1, 144, 571, 6336, 14824, 31529, 31529, 14824, 6336, 571, 144, 1

Offset: 1

Ralf Stephan, Oct 16 2004

There are many versions of this array (or triangle) in the OEIS. This is the main entry, which ideally collects together all the references to the literature and to other versions in the OEIS. But see A004003 for further information. - N. J. A. Sloane, Mar 14 2015

			0,  1,  0,   1,    0,    1, ...
1,  2,  3,   5,    8,   13, ...
0,  3,  0,  11,    0,   41, ...
1,  5, 11,  36,   95,  281, ...
0,  8,  0,  95,    0, 1183, ...
1, 13, 41, 281, 1183, 6728, ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 406-412.
P. E. John, H. Sachs, and H. Zernitz, Problem 5. Domino covers in square chessboards, Zastosowania Matematyki (Applicationes Mathematicae) XIX 3-4 (1987), 635-641.
R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, 2nd ed., pp. 547 and 570.
Darko Veljan, Kombinatorika: s teorijom grafova (Croatian) (Combinatorics with Graph Theory) mentions the value 12988816 = 2^4*901^2 for the 8 X 8 case on page 4.

Alois P. Heinz, Table of n, a(n) for n = 1..1035
M. Aanjaneya and S. P. Pal, Faultfree tromino tilings of rectangles, arXiv:math/0610925 [math.CO], 2006.
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
F. Ardila and R. P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, Journal of Combinatorial Theory, Series A, Volume 77, Issue 1, January 1997, Pages 67-97.
Henry Cohn, 2-adic behavior of numbers of domino tilings, arXiv:math/0008222 [math.CO], 2000.
Henry Cohn, 2-adic behavior of numbers of domino tilings, Electronic Journal of Combinatorics, 6 (1999), #R14.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Steven R. Finch, The Dimer Problem [From Steven Finch, Apr 20 2019]
Steven R. Finch, Two Dimensional Monomer Dimer Constant [Broken link]
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 363.
Laura Florescu, Daniela Morar, David Perkinson, Nicholas Salter, and Tianyuan Xu, Sandpiles and Dominos, Electronic Journal of Combinatorics, Volume 22(1), 2015.
W. Jockusch, Perfect matchings and perfect squares J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
Peter E. John and Horst Sachs, On a strange observation in the theory of the dimer problem, arXiv:math/9801094 [math.CO], 1998.
Peter E. John and Horst Sachs, On a strange observation in the theory of the dimer problem, Discrete Math. 216 (2000), no. 1-3, 211-219.
Yuhi Kamio, Junnosuke Koizumi, and Toshihiko Nakazawa, Quadratic residues and domino tilings, arXiv:2311.13597 [math.NT], 2023.
David Klarner and Jordan Pollack, Domino tilings of rectangles with fixed width, Disc. Math. 32 (1980) 45-52, Table 1.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
P. W. Kasteleyn, The statistics of dimers on a lattice, I. the number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209-1225.
Douglas M. McKenna, The Art of Space-Filling Domino Curves, Bridges Conference Proceedings, 2024, pp. 319-326.
L. Pachter, Combinatorial approaches and conjectures for 2-divisibility problems concerning domino tilings of polyominoes, Electronic Journal of Combinatorics 4 (1997), #R29.
J. Propp, Dimers and Dominoes
J. Propp, Enumeration of Matchings: Problems and Progress, arXiv:math/9904150v2 [math.CO], 1999.
Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9:3 (2005), 609-640.
R. C. Read, A Note on Tiling Rectangles with Dominoes, The Fibonacci Quarterly, 18.1 (1980), 24-27.
R. P. Stanley, A combinatorial miscellany
H. N. V. Temperley and Michael E. Fisher, Dimer problem in statistical mechanics -- an exact result, Philos. Mag. (8) 6 (1961), 1061-1063.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Domino Tiling
Eric Weisstein, Illustration for T(4,4) = 36, from Domino Tilings web page (see previous link) [Included with permission]
Index entries for sequences related to dominoes

See A187596 for another version (with m >= 0, n >= 0). See A187616 for a triangular version. See also A187617, A187618.
See also A004003 for more literature on the dimer problem.
Rows 2-13, 16 (without zeros) are A000045, A001835, A005178, A003775, A028468, A028469, A028470, A028471, A028472, A028473, A028474, A241908, A340532.
Main diagonal is A004003.
Cf. A103997, A103999, A233320, A230031, A233427.

(Maple code for the even-numbered rows from N. J. A. Sloane, Mar 15 2015. This is not totally satisfactory since it uses floating point. However, it is useful for getting the initial values quickly.)
Digits:=100;
p:=evalf(Pi);
z:=proc(h,d) global p; evalf(cos( h*p/(2*d+1) )); end;
T:=proc(m,n) global z; round(mul( mul( 4*z(h,m)^2+4*z(k,n)^2, k=1..n), h=1..m)); end;
[seq(T(1,n),n=0..10)]; # A001519
[seq(T(2,n),n=0..10)]; # A188899
[seq(T(3,n),n=0..10)]; # A256044
[seq(T(n,n),n=0..10)]; # A004003

T[?OddQ, ?OddQ] = 0;
T[m_, n_] := Product[2*(2+Cos[2j*Pi/(m+1)]+Cos[2k*Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}];
Flatten[Table[Round[T[m-n+1, n]], {m, 1, 12}, {n, 1, m}]] (* Jean-François Alcover, Nov 25 2011, updated May 28 2022 *)

{T(n, k) = sqrtint(abs(polresultant(polchebyshev(n, 2, x/2), polchebyshev(k, 2, I*x/2))))} \\ Seiichi Manyama, Apr 13 2020

T(m, n) = Product_{j=1..ceiling(m/2)} Product_{k=1..ceiling(n/2)} (4*cos(j*Pi/(m+1))^2 + 4*cos(k*Pi/(n+1))^2).

Old link fixed and new link added by Frans J. Faase, Feb 04 2009

Entry edited by N. J. A. Sloane, Mar 15 2015

A005178 Number of domino tilings of 4 X (n-1) board.

Original entry on oeis.org

0, 1, 1, 5, 11, 36, 95, 281, 781, 2245, 6336, 18061, 51205, 145601, 413351, 1174500, 3335651, 9475901, 26915305, 76455961, 217172736, 616891945, 1752296281, 4977472781, 14138673395, 40161441636, 114079985111, 324048393905

Offset: 0

N. J. A. Sloane, David Singmaster, Frans J. Faase

Or, number of perfect matchings in graph P_4 X P_{n-1}.
a(0) = 0, a(1) = 1 by convention.
It is easy to see that the g.f. for indecomposable tilings, i.e., those that cannot be split vertically into smaller tilings, is g = x + 4x^2 + 2x^3 + 3x^4 + 2x^5 + 3x^6 + 2x^7 + 3x^8 + ... = x + 4x^2 + x^3*(2+3x)/(1-x^2); then g.f. = 1/(1-g) = (1-x^2)/(1-x-5x^2-x^3+x^4). - Emeric Deutsch, Oct 16 2006
This is a divisibility sequence; that is, if n divides m, then a(n) divides a(m). - T. D. Noe, Dec 22 2008
From Artur Jasinski, Dec 20 2008: (Start)
All numbers in this sequence are:
congruent to 0 mod 100 if n is congruent to 14 or 29 mod 30
congruent to 1 mod 100 if n is congruent to 0 or 1 or 12 or 16 or 27 or 28 mod 30
congruent to 5 mod 100 if n is congruent to 2 or 11 or 17 or 26 mod 30
congruent to 11 mod 100 if n is congruent to 3 or 25 mod 30
congruent to 36 mod 100 if n is congruent to 4 or 9 or 19 or 24 mod 30
congruent to 45 mod 100 if n is congruent to 8 or 20 mod 30
congruent to 51 mod 100 if n is congruent to 13 or 15 mod 30
congruent to 61 mod 100 if n is congruent to 10 or 18 mod 30
congruent to 81 mod 100 if n is congruent to 6 or 7 or 21 or 22 mod 30
congruent to 95 mod 100 if n is congruent to 5 or 23 mod 30
(End)
This is the case P1 = 1, P2 = -7, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

			For n=2 the graph is
  o-o-o-o
and there is one perfect tiling:
  o-o o-o
For n=3 the graph is
  o-o-o-o
  | | | |
  o-o-o-o
and there are five perfect tilings:
  o o o o
  | | | |
  o o o o
two like:
  o o o-o
  | | ...
  o o o-o
and this
  o-o o-o
  .......
  o-o o-o
and this
  o o-o o
  | ... |
  o o-o o
a(n+1)=r(n)-r(n-2), r(n)=if n=0 then 1 else sum(sum(binomial(k,j)*sum(binomial(j,i-j)*5^(i-j)*binomial(k-j,n-i-3*(k-j))*(-1)^(n-i-3*(k-j)),i,j,n-k+j),j,0,k),k,1,n), n>1. - _Vladimir Kruchinin_, Sep 08 2010

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, p. 292.

T. D. Noe, Table of n, a(n) for n = 0..200
Steve Butler, Jason Ekstrand, and Steven Osborne, Counting Tilings by Taking Walks in a Graph, A Project-Based Guide to Undergraduate Research in Mathematics, Birkhäuser, Cham (2020), see page 158.
Curtis Cooper and Robert E. Kennedy, Problem B-735: Soln 1, Problem B-735: Soln 2, Square Root of a Recurrence, The Fibonacci Quarterly, 32(2):185-186, May 1994, and 32(4):374-375, Aug 1994.
F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Preliminary version of paper that appeared in Ars Combin. 49 (1998), 129-154.
F. Faase, Counting Hamiltonian cycles in product graphs
F. Faase, Results from the counting program
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
R. J. Mathar, Paving Rectangular Regions with Rectangular Tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Simone Rinaldi and D. G. Rogers, Indecomposability: polyominoes and polyomino tilings, The Mathematical Gazette 92.524 (2008): 193-204.
David Singmaster, Letter to N. J. A. Sloane, Oct 3 1982.
Thotsaporn "Aek" Thanatipanonda, Statistics of Domino Tilings on a Rectangular Board, Fibonacci Quart. 57 (2019), no. 5, 145-153. See p. 151.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Li Zhou, Northwestern University Math Problem Solving Group, Christopher Carl Heckman and Jerry Minkus, Tiling 4-Rowed Rectangles with Dominoes: 11187, The American Mathematical Monthly 114 (2007): 554-556.
Index to divisibility sequences
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (1,5,1,-1).

Row 4 of array A099390.
For all matchings see A033507.
Cf. A003775, A028468, A028469, A028470.
Cf. A003757. - T. D. Noe, Dec 22 2008
Bisection (odd part) gives A188899. - Alois P. Heinz, Oct 28 2012
Column k=2 of A250662.

a[0]:=1: a[1]:=1: a[2]:=5: a[3]:=11: for n from 4 to 26 do a[n]:=a[n-1]+5*a[n-2]+a[n-3]-a[n-4] od: seq(a[n],n=0..26); # Emeric Deutsch, Oct 16 2006
A005178:=-(-1-4*z-z**2+z**3)/(1-z-5*z**2-z**3+z**4) # conjectured (correctly) by Simon Plouffe in his 1992 dissertation; gives sequence apart from an initial 1

CoefficientList[Series[x(1-x^2)/(1-x-5x^2-x^3+x^4), {x,0,30}], x] (* T. D. Noe, Dec 22 2008 *)
LinearRecurrence[{1, 5, 1, -1}, {0, 1, 1, 5}, 28] (* Robert G. Wilson v, Aug 08 2011 *)
a[0] = 0; a[n_] := Product[2(2+Cos[2j Pi/5]+Cos[2k Pi/n]), {k, 1, (n-1)/2}, {j, 1, 2}] // Round;
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 20 2018 *)

r(n):=if n=0 then 1 else sum(sum(binomial(k,j)*sum(binomial(j,i-j)*5^(i-j)*binomial(k-j,n-i-3*(k-j))*(-1)^(n-i-3*(k-j)),i,j,n-k+j),j,0,k),k,1,n); a(n):=r(n)-r(n-2); /* Vladimir Kruchinin, Sep 08 2010 */

a(n) = a(n-1) + 5*a(n-2) + a(n-3) - a(n-4).
G.f.: x*(1 - x^2)/(1 - x - 5*x^2 - x^3 + x^4).
Limit_{n->oo} a(n)/a(n-1) = (1 + sqrt(29) + sqrt(14 + 2*sqrt(29)))/4 = 2.84053619409... - Philippe Deléham, Jun 12 2005
a(n) = (5*sqrt(29)/145)*(((1+sqrt(29)+sqrt(14+2*sqrt(29)))/4)^n+((1+sqrt(29)-sqrt(14+2*sqrt(29)))/4)^n-((1-sqrt(29)+sqrt(14-2*sqrt(29)))/4)^n-((1-sqrt(29)-sqrt(14-2*sqrt(29)))/4)^n). - Tim Monahan, Jul 30 2011
From Peter Bala, Mar 31 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(29))/4 and beta = (1 - sqrt(29))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 7/4; 1, 1/2].
a(n) = U(n-1,i*(1 + sqrt(5))/4)*U(n-1,i*(1 - sqrt(5))/4), where U(n,x) denotes the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)
a(n) = A129113(n+2) - A129113(n). - R. J. Mathar, May 03 2021

Amalgamated with (former) A003692, Dec 30 1995
Name changed and 0 prepended by T. D. Noe, Dec 22 2008
Edited by N. J. A. Sloane, Nov 15 2009

A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 19, 281, 1, 1, 1, 1, 1, 1, 57, 1183, 1, 1, 1, 1, 1, 1, 26, 121, 6728, 1, 1, 1, 1, 1, 1, 1, 75, 783, 31529, 1, 1, 1, 1, 1, 1, 1, 34, 154, 2861, 167089, 1, 1, 1, 1, 1, 1, 1, 1, 95, 269, 8133, 817991, 1, 1

Offset: 0

Alois P. Heinz, Nov 29 2014

A(n,n) = A034856(n+2) for n>=2.

			Square array A(n,k) begins:
  1, 1,      1,    1,    1,   1,   1,   1,  1, ...
  1, 1,      1,    1,    1,   1,   1,   1,  1, ...
  1, 1,     13,    1,    1,   1,   1,   1,  1, ...
  1, 1,     41,   19,    1,   1,   1,   1,  1, ...
  1, 1,    281,   57,   26,   1,   1,   1,  1, ...
  1, 1,   1183,  121,   75,  34,   1,   1,  1, ...
  1, 1,   6728,  783,  154,  95,  43,   1,  1, ...
  1, 1,  31529, 2861,  269, 190, 117,  53,  1, ...
  1, 1, 167089, 8133, 1732, 325, 229, 141, 64, ...

Alois P. Heinz, Antidiagonals n = 0..35, flattened
Wikipedia, Polyomino

Columns k=0+1,2-10 give: A000012, A028468, A251073, A251074, A247218, A251075, A251076, A251077, A251078, A251079.

Cf. A034856, A250662.

b:= proc(n, l) option remember; local d, k; d:= nops(l)/3;
      if n=0 then 1
    elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
    else for k while l[k]>0 do od;
         `if`(n2*d+1 or max(l[k..k+d-1][])>0, 0,
          b(n, [l[1..k-1][], 1$d, l[k+d..3*d][]]))
      fi
    end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$3*k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/3, k}, Which[n == 0, 1,  Min[l] > 0, Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0 , k++]; If[n d]]] + If[d == 1 || k > 2d + 1 || Max[l[[k ;; k + d - 1]]] > 0,  0,  b[n, Join[l[[1 ;; k-1]], Array[1&, d],  l[[k+d ;; 3*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 3k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)

A189006 Array A(m,n) read by antidiagonals: number of domino tilings of the m X n grid with upper left corner removed iff m*n is odd, (m>=0, n>=0).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 4, 5, 1, 1, 1, 1, 8, 11, 11, 8, 1, 1, 1, 1, 13, 15, 36, 15, 13, 1, 1, 1, 1, 21, 41, 95, 95, 41, 21, 1, 1, 1, 1, 34, 56, 281, 192, 281, 56, 34, 1, 1, 1, 1, 55, 153, 781, 1183, 1183, 781, 153, 55, 1, 1, 1, 1, 89, 209, 2245, 2415, 6728, 2415, 2245, 209, 89, 1, 1

Offset: 0

Alois P. Heinz, Apr 15 2011

			A(3,3) = 4, because there are 4 domino tilings of the 3 X 3 grid with upper left corner removed:
  . .___. . .___. . .___. . .___.
  ._|___| ._|___| ._| | | ._|___|
  | |___| | | | | | |_|_| |___| |
  |_|___| |_|_|_| |_|___| |___|_|
Array begins:
  1, 1,  1,  1,   1,    1,    1, ...
  1, 1,  1,  1,   1,    1,    1, ...
  1, 1,  2,  3,   5,    8,   13, ...
  1, 1,  3,  4,  11,   15,   41, ...
  1, 1,  5, 11,  36,   95,  281, ...
  1, 1,  8, 15,  95,  192, 1183, ...
  1, 1, 13, 41, 281, 1183, 6728, ...

Alois P. Heinz, Antidiagonals n = 0..75, flattened
Eric Weisstein's World of Mathematics, Perfect Matching
Wikipedia, FKT algorithm
Wikipedia, Matching (graph theory)
Index entries for sequences related to dominoes

Rows m=0+1, 2-12 give: A000012, A000045(n+1), A002530(n+1), A005178(n+1), A189003, A028468, A189004, A028470, A189005, A028472, A210724, A028474.
Main diagonal gives: A189002.
Cf. A099390, A187596, A187616, A187617, A187618, A004003.

with(LinearAlgebra):
A:= proc(m, n) option remember; local i, j, s, t, M;
      if m=0 or n=0 then 1
    elif m1 or j>1 or s=0 then
               if j

A[1, 1] = 1; A[m_, n_] := A[m, n] = Module[{i, j, s, t, M}, Which[m == 0 || n == 0, 1, m < n, A[n, m], True, s = Mod[n*m, 2];M[i_, j_] /; j < i := -M[j, i]; M[, ] = 0; For[i = 1, i <= n, i++, For[j = 1, j <= m, j++, t = (i-1)*m+j-s; If[i > 1 || j > 1 || s == 0, If[j < m, M[t, t+1] = 1]; If[i < n, M[t, t+m] = 1-2*Mod[j, 2]]]]]; Sqrt[Det[Array[M, {n*m-s, n*m-s}]]]]]; Table[Table[A[m, d-m], {m, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 26 2013, translated from Maple *)

A028471 Number of perfect matchings (or domino tilings) in the graph P_9 X P_2n.

Original entry on oeis.org

1, 55, 6336, 817991, 108435745, 14479521761, 1937528668711, 259423766712000, 34741645659770711, 4652799879944138561, 623139489426439754945, 83456125990631342400791, 11177167872295392172767936, 1496943834332592837945956455, 200483802581126644843760725601

Offset: 0

Per H. Lundow

nonn

Sean A. Irvine, Table of n, a(n) for n = 0..150
J. L. Hock and R. B. McQuistan, A note on the occupational degeneracy for dimers on a saturated two-dimensional lattice space, Discrete Applied Mathematics, 1984, v.8, 101-104.
Per Hakan Lundow, Computation of matching polynomials and the number of 1-factors in polygraphs, Research report, No 12, 1996, Department of Math., Umea University, Sweden.
Per Hakan Lundow, Enumeration of matchings in polygraphs, 1998.
Index entries for linear recurrences with constant coefficients, signature (209, -11936, 274208, -3112032, 19456019, -70651107, 152325888, -196664896, 152325888, -70651107, 19456019, -3112032, 274208, -11936, 209, -1).

Cf. A000045, A001835, A005178, A003775, A028468, A028469, A028470.

Row 9 of array A099390.

T[?OddQ, ?OddQ] = 0;
T[m_, n_] := Product[2(2+Cos[2 j Pi/(m+1)]+Cos[2 k Pi/(n+1)]), {k, 1, n/2}, {j, 1, m/2}];
a[n_] := T[2n, 9] // Round;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 28 2022 *)

{a(n) = sqrtint(polresultant(polchebyshev(2*n, 2, x/2), polchebyshev(9, 2, I*x/2)))} \\ Seiichi Manyama, Apr 13 2020

a(n) = 209*a(n-1) - 11936*a(n-2) + 274208*a(n-3) - 3112032*a(n-4) + 19456019*a(n-5) - 70651107*a(n-6) + 152325888*a(n-7) - 196664896*a(n-8) + 152325888*a(n-9) - 70651107*a(n-10) + 19456019*a(n-11) - 3112032*a(n-12) + 274208*a(n-13) - 11936*a(n-14) + 209*a(n-15) - a(n-16). - Jay Anderson (horndude77(AT)gmail.com), Apr 07 2007

G.f.: (1 - 154x + 6777x^2 - 123961x^3 + 1132714x^4 - 5684515x^5 + 16401668x^6 - 27757938x^7 + 27757938*x^8 - 16401668x^9 + 5684515x^10 - 1132714x^11 + 123961x^12 -6777x^13 + 154x^14 - x^15)/(1 - 209x + 11936x^2 - 274208x^3 + 3112032x^4 - 19456019x^5 + 70651107x^6 - 152325888x^7 + 196664896x^8 - 152325888x^9 + 70651107x^10 -19456019x^11 + 3112032x^12 - 274208x^13 + 11936x^14 - 209x^15 + x^16). - Sergey Perepechko, Nov 23 2012

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A256044 6th row of array in A099390.

Original entry on oeis.org

1, 13, 281, 6728, 167089, 4213133, 106912793, 2720246633, 69289288909, 1765722581057, 45005025662792, 1147185247901449, 29242880940226381, 745439797095329713, 19002353776441540177, 484398978524471931341, 12348080425980866090537, 314771823879840325570888

Offset: 0

N. J. A. Sloane, Mar 14 2015

Alois P. Heinz, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (40,-416,1224,-1224,416,-40,1).

Cf. A099390.

Bisection (even part) of A028468 and 6th row of A187596. - Alois P. Heinz, Mar 16 2015

I:=[1,13,281,6728,167089,4213133,106912793]; [n le 7 select I[n] else 40*Self(n-1)-416*Self(n-2)+1224*Self(n-3)-1224*Self(n-4)+416*Self(n-5)-40*Self(n-6)+Self(n-7): n in [1..30]]; // Vincenzo Librandi, Aug 20 2018

a[n_] := Product[2(2+Cos[2j Pi/7]+Cos[2k Pi/(2n+1)]), {k, 1, n}, {j, 1, 3}] // Round;
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 20 2018 *)
LinearRecurrence[{40, -416, 1224, -1224, 416, -40, 1}, {1, 13, 281, 6728, 167089, 4213133, 106912793}, 20] (* Vincenzo Librandi, Aug 20 2018 *)

x='x+O('x^100); Vec(-(x^6-27*x^5+177*x^4-328*x^3+177*x^2-27*x+1)/((x-1)*(x^3-26*x^2+13*x-1)*(x^3-13*x^2+26*x-1))) \\ Altug Alkan, Mar 23 2016

G.f.: -(x^6 - 27*x^5 + 177*x^4 - 328*x^3 + 177*x^2 - 27*x + 1) / ((x - 1)*(x^3 - 26*x^2 + 13*x - 1)*(x^3 - 13*x^2 + 26*x - 1)). - Alois P. Heinz, Mar 16 2015

a(n) = 40*a(n-1) - 416*a(n-2) + 1224*a(n-3) - 1224*a(n-4) + 416*a(n-5) - 40*a(n-6) + a(n-7). - Vincenzo Librandi, Aug 20 2018

a(8)-a(17) from Alois P. Heinz, Mar 16 2015

cp's OEIS Frontend

A099390 Array T(m,n) read by antidiagonals: number of domino tilings (or dimer tilings) of the m X n grid (or m X n rectangle), for m>=1, n>=1.

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A005178 Number of domino tilings of 4 X (n-1) board.

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A251072 Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A189006 Array A(m,n) read by antidiagonals: number of domino tilings of the m X n grid with upper left corner removed iff m*n is odd, (m>=0, n>=0).

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A028471 Number of perfect matchings (or domino tilings) in the graph P_9 X P_2n.

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A256044 6th row of array in A099390.

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