A028469 -id:A028469 - 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

A003696 Number of spanning trees in P_4 X P_n.

Original entry on oeis.org

1, 56, 2415, 100352, 4140081, 170537640, 7022359583, 289143013376, 11905151192865, 490179860527896, 20182531537581071, 830989874753525760, 34214941811800329425, 1408756312731277540744

Offset: 1

Frans J. Faase

Also number of domino tilings of the 7 X (2n-1) rectangle with upper left corner removed. - Alois P. Heinz, Apr 14 2011

A linear divisibility sequence of order 8; a(n) divides a(m) whenever n divides m. It is the product of a 2nd-order Lucas sequence and a 4th-order linear divisibility sequence. - Peter Bala, Apr 27 2014

F. Faase, On the number of specific spanning subgraphs of the graphs G X P_n, Ars Combin. 49 (1998), 129-154.

T. D. Noe, Table of n, a(n) for n = 1..200
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
P. Raff, Spanning Trees in Grid Graphs, arXiv:0809.2551 [math.CO], 2008.
P. Raff, Analysis of the Number of Spanning Trees of P_4 x P_n. Contains sequence, recurrence, generating function, and more.
Roettger, E. L.; Williams, H. C.; Guy, R. K., Some extensions of the Lucas functions, Springer Proceedings in Mathematics & Statistics 43, 271-311 (2013), chapter 5.
Index to divisibility sequences
Index entries for sequences related to trees
Index entries for linear recurrences with constant coefficients, signature (56, -672, 2632, -4094, 2632, -672, 56, -1).

A row of A116469. - N. J. A. Sloane, May 27 2012
Bisection of A189004. - Alois P. Heinz, Sep 20 2012
A001353, A161158, A159764.

seq(resultant(simplify(ChebyshevU(3, (x-4)*(1/2))), simplify(ChebyshevU(n-1, (1/2)*x)), x), n = 1 .. 14); # Peter Bala, Apr 27 2014

LinearRecurrence[{56, -672, 2632, -4094, 2632, -672, 56, -1}, {1, 56, 2415, 100352, 4140081, 170537640, 7022359583, 289143013376}, 20] (* Jean-François Alcover, Feb 28 2020 *)

{a(n) = polresultant((x-4)*(x^2-8*x+14), polchebyshev(n-1, 2, x/2))}; /* Michael Somos, Oct 31 2022 */

a(1) = 1,
a(2) = 56,
a(3) = 2415,
a(4) = 100352,
a(5) = 4140081,
a(6) = 170537640,
a(7) = 7022359583,
a(8) = 289143013376 and
a(n) = 56a(n-1) - 672a(n-2) + 2632a(n-3) - 4094a(n-4) + 2632a(n-5) - 672a(n-6) + 56a(n-7) - a(n-8).
G.f.: x(x^6-49x^4+112x^3-49x^2+1) / (x^8-56x^7 +672x^6-2632x^5 +4094x^4 -2632x^3 +672x^2-56x+1). - Paul Raff, Mar 06 2009
From Peter Bala, Apr 27 2014: (Start)
a(n) = Resultant( U(3,(x-4)/2),U(n-1,x/2) ), where U(n,x) denotes the Chebyshev polynomial of the second kind. The polynomial U(3,(x-4)/2) = x^3 - 12*x^2 + 46*x - 56 (see A159764) has zeros z_1 = 4, z_2 = 4 + sqrt(2) and z_3 = 4 - sqrt(2). Hence a(n) = U(n-1,2)*U(n-1,1/2*(4 + sqrt(2)))*U(n-1,1/2*(4 - sqrt(2))).
a(n) = A001353(n)*A161158(n-1). (End)
a(n) = (9/3968)*(A028469(n+3)-A028469(n-4)) - (497/3968)*(A028469(n+2)-A028469(n-3)) + (5687/3968)*(A028469(n+1)-A028469(n-2)) - (19983/3968)*(A028469(n)-A028469(n-1)), n>3. - Sergey Perepechko, May 02 2016
a(n) = -a(-n) for all n in Z. - Michael Somos, Oct 31 2022

Added recurrence from Faase's web page. - N. J. A. Sloane, Feb 03 2009

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

A189004 Number of domino tilings of the 7 X n grid with upper left corner removed iff n is odd.

Original entry on oeis.org

1, 1, 21, 56, 781, 2415, 31529, 100352, 1292697, 4140081, 53175517, 170537640, 2188978117, 7022359583, 90124167441, 289143013376, 3710708201969, 11905151192865, 152783289861989, 490179860527896, 6290652543875133

Offset: 0

Alois P. Heinz, Apr 15 2011

Alois P. Heinz, Table of n, a(n) for n = 0..400
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (0, 56, 0, -672, 0, 2632, 0, -4094, 0, 2632, 0, -672, 0, 56, 0, -1).

7th row of array A189006.

Bisection gives: A028469 (even part), A003696 (odd part).

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}]] ]]];
a[n_] := A[7, n];
a /@ Range[0, 20] (* Jean-François Alcover, Feb 27 2020, after Alois P. Heinz in A189006 *)

G.f.: -(x^14-x^13-35*x^12+277*x^10 +49*x^9-727*x^8 -112*x^7+727*x^6 +49*x^5-277*x^4 +35*x^2-x-1) / (x^16-56*x^14 +672*x^12-2632*x^10 +4094*x^8-2632*x^6 +672*x^4-56*x^2+1).

A360799 Numbers m with m mod 3 = q, q != 2, such that the number of ones in its base-2 representation is even if q=0 and odd if q=1.

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 12, 13, 15, 16, 18, 19, 22, 24, 25, 27, 28, 30, 31, 33, 36, 37, 39, 45, 48, 49, 51, 52, 54, 55, 57, 60, 61, 63, 64, 66, 67, 70, 72, 73, 75, 76, 78, 79, 82, 88, 90, 91, 94, 96, 97, 99, 100, 102, 103, 105, 108, 109, 111, 112, 114, 115, 118, 120, 121

Offset: 0

Gerhard Kirchner, Feb 24 2023

nonn

For q=0, the terms in A180963 are excluded.
The terms of the sequence occur, with some exceptions, while tiling a wall (odd width w) with 1 X 2 dominos. The current tiling status can be described by a number x with 0 <= x < 2^w. In the base-2 representation, 1 stands for an overstanding unit square, see example.
Statement:
The tiling always starts with q=1 and an odd number of ones (type 1) and is followed by a term with q=0 and an even number of ones (type 2) and so on, alternately.
Proof:
Start, provisionally, with w upright dominos. The corresponding term is x = (11..1) = 2^w-1 with x mod 3 = 1 (type 1). Another first profile can be generated by replacing a pair of adjacent upright dominos with one flat domino. In the base-2 representation, this is the subtraction (11..11..1) - (00..11..0) = (11..00..1). The subtrahend is 3*2^j with 0 <= j < w. Therefore, the modified term also is type 1. This way, any first profile can be found and it is type 1.
In the next provisional step, an upright domino is placed on each not overstanding unit square. If p1 is the first profile, then the second is p2 = 2^w - 1 - p1 with p2 mod 3 = 0. Moreover, the transition from p1 to p2 exchanges the ones and zeros such that p2 is type 2. Again, replacing adjacent upright dominos by one flat domino does not change the type of the profile. The next profile is type 1 and so on. QED. Condition to be satisfied by a tiling profile: The continued removal of 00 and 11 (reduction) leads to (0) or (1). Example: a(10)=18=(10010) -> (110) -> (0). The first exceptions are a(314) = 682 = (01010101010), a(611) = 1365 = (10101010101) and a(988) = 2218 = (0100010101010). Note that the reduction of 2218 is 682.

			 5 X 4 wall is tiled bottom-up with 1 X 2 dominos:
                                      _    ___ ___ _
                 _ _          _ _ ___| |  |_ _|___| |
        _       | | |_ ___   | | |_ _|_|  | | |_ _|_|
    ___| |___   |_|_| |___|  |_|_| |___|  |_|_| |___|
   |___|_|___|  |___|_|___|  |___|_|___|  |___|_|___|
    0 0 1 0 0    1 1 0 0 0    0 0 0 0 1    0 0 0 0 0
     4 = a(3)   24 = a(14)     1 = a(1)     0 = a(0)

Cf. A001835, A003775, A028469, A028471, A180963, A187616, A360800.

block(kmax: 100, a:[],
 even_ones(x):= block(su:0,
  while x>0 do(p: mod(x,2), x:(x-p)/2, su:su+p),
   return(mod(su,2))),
for k from 0 thru kmax do(r:mod(k,3),
 if r<2 and r=even_ones(k) then a:append(a,[k])),a);

isok(m) = my(k=m%3); if (hammingweight(m) % 2, k==1, k==0); \\ Michel Marcus, Feb 27 2023

A360800 Numbers Sum_{i=1..2r+1} 2^k(i) such that k(1) is even and, for r > 0 and i < 2r+1, the difference k(i+1)-k(i) is > 0 and odd.

Original entry on oeis.org

1, 4, 7, 16, 19, 25, 28, 31, 64, 67, 73, 76, 79, 97, 100, 103, 112, 115, 121, 124, 127, 256, 259, 265, 268, 271, 289, 292, 295, 304, 307, 313, 316, 319, 385, 388, 391, 400, 403, 409, 412, 415, 448, 451, 457, 460, 463, 481, 484, 487, 496, 499, 505, 508, 511, 1024

Offset: 1

Gerhard Kirchner, Feb 24 2023

nonn

This is a subsequence of A360799. Another description of the terms: in the base-2 representation, the number of ones is odd and all zeros are grouped in blocks of even length.

That is why the terms less than 2^(2j+1) describe start profiles for tiling a (2j+1) X m wall with 1 X 2 dominos, see examples and A360799.

			A 5 X m wall is tiled bottom-up with dominos, start profiles:
            _        _            _ _ _    _     _ _    _ _ _ _ _
    ___ ___| |   ___| |___    ___| | | |  | |___| | |  | | | | | |
   |___|___|_|  |___|_|___|  |___|_|_|_|  |_|___|_|_|  |_|_|_|_|_|
    0 0 0 0 1    0 0 1 0 0    0 0 1 1 1    1 0 0 1 1    1 1 1 1 1
    1 = a(1)     4 = a(2)     7 = a(3)     19 = a(5)    31 = a(7)
    also the mirror images of 1 (16), 19 (25) and 7 (28).

Cf. A001835, A003775, A028469, A028471, A180963, A187616, A360799.

block(kmax: 100, a:[],
 oddsum(y):= block(su1:0, su2:0, pold:0, ok: true,
  while y>0 and ok do(p:mod(y,2), y:(y-p)/2,
   if p=1 then(if pold=0 and su2=1 then ok:false, su1:1-su1, su2:0)
   elseif p=0 then su2:1-su2, pold:p), return(is(ok and su1=1))),
for k from 1 thru kmax do if oddsum(k) then a:append(a,[k]),a);

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

Extensions

A003696 Number of spanning trees in P_4 X P_n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A189004 Number of domino tilings of the 7 X n grid with upper left corner removed iff n is odd.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A360799 Numbers m with m mod 3 = q, q != 2, such that the number of ones in its base-2 representation is even if q=0 and odd if q=1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maxima

PARI

A360800 Numbers Sum_{i=1..2r+1} 2^k(i) such that k(1) is even and, for r > 0 and i < 2r+1, the difference k(i+1)-k(i) is > 0 and odd.

Views

Author

Keywords

Comments