A230031 -id:A230031 - cp's OEIS frontend

A007598 Squared Fibonacci numbers: a(n) = F(n)^2 where F = A000045.

0, 1, 1, 4, 9, 25, 64, 169, 441, 1156, 3025, 7921, 20736, 54289, 142129, 372100, 974169, 2550409, 6677056, 17480761, 45765225, 119814916, 313679521, 821223649, 2149991424, 5628750625, 14736260449, 38580030724, 101003831721, 264431464441, 692290561600

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

a(n)*(-1)^(n+1) = (2*(1-T(n,-3/2))/5), n>=0, with Chebyshev's polynomials T(n,x) of the first kind, is the r=-1 member of the r-family of sequences S_r(n) defined in A092184 where more information can be found. - Wolfdieter Lang, Oct 18 2004
From Giorgio Balzarotti, Mar 11 2009: (Start)
Determinant of power series with alternate signs of gamma matrix with determinant 1!.
a(n) = Determinant(A - A^2 + A^3 - A^4 + A^5 - ... - (-1)^n*A^n) where A is the submatrix A(1..2,1..2) of the matrix with factorial determinant.
A = [[1,1,1,1,1,1,...], [1,2,1,2,1,2,...], [1,2,3,1,2,3,...], [1,2,3,4,1,2,...], [1,2,3,4,5,1,...], [1,2,3,4,5,6,...], ...]; note: Determinant A(1..n,1..n) = (n-1)!.
a(n) is even with respect to signs of power of A.
See A158039...A158050 for sequence with matrix 2!, 3!, ... (End)
Equals the INVERT transform of (1, 3, 2, 2, 2, ...). Example: a(7) = 169 = (1, 1, 4, 9, 25, 64) dot (2, 2, 2, 2, 3, 1) = (2 + 2 + 8 + 18 + 75 + 64). - Gary W. Adamson, Apr 27 2009
This is a divisibility sequence.
a(n+1)*(-1)^n, n>=0, is the sequence of the alternating row sums of the Riordan triangle A158454. - Wolfdieter Lang, Dec 18 2010
a(n+1) is the number of tilings of a 2 X 2n rectangle with n tetrominoes of any shape, cf. A230031. - Alois P. Heinz, Nov 29 2013
This is the case P1 = 1, P2 = -6, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014
Differences between successive golden rectangle numbers A001654. - Jonathan Sondow, Nov 05 2015
a(n+1) is the number of 2 X n matrices that can be obtained from a 2 X n matrix by moving each element to an adjacent position, horizontally or vertically. This is because F(n+1) is the number of domino tilings of that matrix, therefore with a checkerboard coloring and two domino tilings we can move the black element of each domino of the first tiling to the white element of the same domino and similarly move the white element of each domino of the second tiling to the black element of the same domino. - Fabio Visonà, May 04 2022
In general, squaring the terms of a second-order linear recurrence with signature (c,d) will result in a third-order linear recurrence with signature (c^2+d,(c^2+d)*d,-d^3). - Gary Detlefs, Jan 05 2023

			G.f. = x + x^2 + 4*x^3 + 9*x^4 + 25*x^5 + 64*x^6 + 169*x^7 + 441*x^8 + ...

Arthur T. Benjamin and Jennifer J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 8.
Ross Honsberger, Mathematical Gems III, M.A.A., 1985, p. 130.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Richard P. Stanley, Enumerative Combinatorics I, Example 4.7.14, p. 251.

Indranil Ghosh, Table of n, a(n) for n = 0..2389 (terms 0..200 from T. D. Noe)
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.
Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 42, 2012, pp. 2053-2059.
R. C. Alperin, A family of nonlinear recurrences and their linear solutions, Fib. Q., 57:4 (2019), 318-321.
Paul S. Bruckman, Problem B-1023: And a cubic as a sum of two squares, Fibonacci Quarterly, Vol. 45, Number 2; May 2007; p. 186.
Andrej Dujella, A bijective proof of Riordan's theorem on powers of Fibonacci numbers, Discrete Math. 199 (1999), no. 1-3, 217--220. MR1675924 (99k:05016).
Sergio Falcon, Some series of reciprocal k-Fibonacci numbers, Asian Journal of Mathematics and Computer Research, Vol. 11, No. 3 (2016), pp. 184-191; ResearchGate link.
Dominique Foata and Guo-Niu Han, Nombres de Fibonacci et polynômes orthogonaux, in: Marcello Morelli and Marco Tangheroni (eds.), Leonardo Fibonacci : il tempo, le opere, l'ereditá scientifica, Pisa, 23-25 Marzo 1994, Pisa, Pacini Editore, 1994, pp. 179-200.
Svenja Huntemann and Neil A. McKay, Counting Domineering Positions, arXiv:1909.12419 [math.CO], 2019.
Jong Hyun Kim, Hadamard products and tilings, JIS 12 (2009) 09.7.4.
Toufik Mansour, A note on sum of k-th power of Horadam's sequence, arXiv:math/0302015 [math.CO], 2003.
Toufik Mansour, Squaring the terms of an l-th order linear recurrence, arXiv:math/0303138 [math.CO], 2003.
Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop, and Abiodun A. Opanuga, Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp 4623-4627.
Pantelimon Stanica, Generating functions, weighted and non-weighted sums of powers of second-order recurrence sequences, arXiv:math/0010149 [math.CO], 2000.
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.
Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
Index entries for two-way infinite sequences.
Index to divisibility sequences.

Cf. A000032, A000045, A001254, A001622, A001654, A047946, A056570, A059260, A061646, A065885, A079291, A080097, A105393.
Bisection of A006498 and A074677. First differences of A001654.
Second row of array A103323.
Half of A175395.

List([0..30], n -> Fibonacci(n)^2); # G. C. Greubel, Dec 10 2018

a007598 = (^ 2) . a000045  -- Reinhard Zumkeller, Sep 01 2013

[Fibonacci(n)^2: n in [0..30]]; // Vincenzo Librandi, Apr 14 2011

with(combinat): seq(fibonacci(n)^2, n=0..27); # Zerinvary Lajos, Sep 21 2007

f[n_] := Fibonacci[n]^2; Array[f, 4!, 0] (* Vladimir Joseph Stephan Orlovsky, Oct 25 2009 *)
LinearRecurrence[{2,2,-1},{0,1,1},41] (* Harvey P. Dale, May 18 2011 *)

PARI
```
{a(n) = fibonacci(n)^2};
```

concat(0, Vec(x*(1-x)/((1+x)*(1-3*x+x^2)) + O(x^30))) \\ Altug Alkan, Nov 06 2015

from sympy import fibonacci
def A007598(n): return fibonacci(n)**2 # Chai Wah Wu, Apr 14 2025

[(fibonacci(n))^2 for n in range(0, 28)]# Zerinvary Lajos, May 15 2009

[fibonacci(n)^2 for n in range(30)] # G. C. Greubel, Dec 10 2018

G.f.: x*(1-x)/((1+x)*(1-3*x+x^2)).
a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3), n > 2. a(0)=0, a(1)=1, a(2)=1.
a(-n) = a(n) for all n in Z.
a(n) = A080097(n-2) + 1.
L.g.f.: 1/5*log((1+3*x+x^2)/(1-6*x+x^2)) = Sum_{n>=0} a(n)/n*x^n; special case of l.g.f. given in A079291. - Joerg Arndt, Apr 13 2011
a(0) = 0, a(1) = 1; a(n) = a(n-1) + Sum(a(n-i)) + k, 0 <= i < n where k = 1 when n is odd, or k = -1 when n is even. E.g., a(2) = 1 = 1 + (1 + 1 + 0) - 1, a(3) = 4 = 1 + (1 + 1 + 0) + 1, a(4) = 9 = 4 + (4 + 1 + 1 + 0) - 1, a(5) = 25 = 9 + (9 + 4 + 1 + 1 + 0) + 1. - Sadrul Habib Chowdhury (adil040(AT)yahoo.com), Mar 02 2004
a(n) = (2*Fibonacci(2*n+1) - Fibonacci(2*n) - 2*(-1)^n)/5. - Ralf Stephan, May 14 2004
a(n) = F(n-1)*F(n+1) - (-1)^n = A059929(n-1) - A033999(n).
Sum_{j=0..2*n} binomial(2*n,j)*a(j) = 5^(n-1)*A005248(n+1) for n >= 1 [P. Stanica]. Sum_{j=0..2*n+1} binomial(2*n+1,j)*a(j) = 5^n*A001519(n+1) [P. Stanica]. - R. J. Mathar, Oct 16 2006
a(n) = (A005248(n) - 2*(-1)^n)/5. - R. J. Mathar, Sep 12 2010
a(n) = (-1)^k*(Fibonacci(n+k)^2-Fibonacci(k)*Fibonacci(2*n+k)), for any k. - Gary Detlefs, Dec 13 2010
a(n) = 3*a(n-1) - a(n-2) + 2*(-1)^(n+1), n > 1. - Gary Detlefs, Dec 20 2010
a(n) = Fibonacci(2*n-2) + a(n-2). - Gary Detlefs, Dec 20 2010
a(n) = (Fibonacci(3*n) - 3*(-1)^n*Fibonacci(n))/(5*Fibonacci(n)), n > 0. - Gary Detlefs, Dec 20 2010
a(n) = (Fibonacci(n)*Fibonacci(n+4) - 3*Fibonacci(n)*Fibonacci(n+1))/2. - Gary Detlefs, Jan 17 2011
a(n) = (((3+sqrt(5))/2)^n + ((3-sqrt(5))/2)^n - 2*(-1)^n)/5; without leading zero we would have a(n) = ((3+sqrt(5))*((3+sqrt(5))/2)^n + (3-sqrt(5))*((3-sqrt(5))/2)^n + 4*(-1)^n)/10. - Tim Monahan, Jul 17 2011
E.g.f.: (exp((phi+1)*x) + exp((2-phi)*x) - 2*exp(-x))/5, with the golden section phi:=(1+sqrt(5))/2. From the Binet-de Moivre formula for F(n). - Wolfdieter Lang, Jan 13 2012
Starting with "1" = triangle A059260 * the Fibonacci sequence as a vector. - Gary W. Adamson, Mar 06 2012
a(0) = 0, a(1) = 1; a(n+1) = (a(n)^(1/2) + a(n-1)^(1/2))^2. - Thomas Ordowski, Jan 06 2013
a(n) + a(n-1) = A001519(n), n > 0. - R. J. Mathar, Mar 19 2014
From Peter Bala, Mar 31 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = 3/2 and beta = -1 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, 3/2; 1, 1/2].
a(n) = U(n-1,i/2)*U(n-1,-i/2), 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) = (F(n+2)*F(n+3) - L(n)*L(n+1))/3 for F = A000045 and L = A000032. - J. M. Bergot, Jun 02 2014
0 = a(n)*(+a(n) - 2*a(n+1) - 2*a(n+2)) + a(n+1)*(+a(n+1) - 2*a(n+2)) + a(n+2)*(+a(n+2)) for all n in Z. - Michael Somos, Jun 03 2014
(F(n)*b(n+2))^2 + (F(n+1)*b(n-1))^2 = F(2*n+1)^3 = A001519(n+1)^3, with b(n) = a(n) + 2*(-1)^n and F(n) = A000045(n) (see Bruckman link). - Michel Marcus, Jan 24 2015
a(n) = 1/4*( a(n-2) - a(n-1) - a(n+1) + a(n+2) ). The same recurrence holds for A001254. - Peter Bala, Aug 18 2015
a(n) = F(n)*F(n+1) - F(n-1)*F(n). - Jonathan Sondow, Nov 05 2015
For n>2, a(n) = F(n-2)*(3*F(n-1) + F(n-3)) + F(2*n-5). Also, for n>2 a(n)=2*F(n-3)*F(n) + F(2*n-3) -(2)*(-1)^n. - J. M. Bergot, Nov 05 2015
a(n) = (F(n+2)^2 + L(n+1)^2) - 2*F(n+2)*L(n+1). - J. M. Bergot, Nov 08 2015
a(n) = F(n+3)^2 - 4*F(n+1)*F(n+2). - J. M. Bergot, Mar 17 2016
a(n) = (F(n-2)*F(n+2) + F(n-1)*F(n+1))/2. - J. M. Bergot, May 25 2017
4*a(n) = L(n+1)*L(n-1) - F(n+2)*F(n-2), where L = A000032. - Bruno Berselli, Sep 27 2017
a(n) = F(n+k)*F(n-k) + (-1)^(n+k)*a(k), for every integer k >= 0. - Federico Provvedi, Dec 10 2018
From Peter Bala, Nov 19 2019: (Start)
Sum_{n >= 3} 1/(a(n) - 1/a(n)) = 4/9.
Sum_{n >= 3} (-1)^n/(a(n) - 1/a(n)) =  (10 - 3*sqrt(5))/18.
Conjecture: Sum_{n >= 1, n != 2*k+1} 1/(a(n) + (-1)^n*a(2*k+1)) = 1/a(4*k+2) for k = 0,1,2,.... (End)
Sum_{n>=1} 1/a(n) = A105393. - Amiram Eldar, Oct 22 2020
Product_{n>=2} (1 + (-1)^n/a(n)) = phi (A001622) (Falcon, 2016, p. 189, eq. (3.1)). - Amiram Eldar, Dec 03 2024

A233427 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 5, 0, 0, 5, 0, 1, 1, 0, 0, 56, 0, 56, 0, 0, 1, 1, 0, 0, 0, 501, 501, 0, 0, 0, 1, 1, 0, 0, 0, 0, 4006, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 27950, 27950, 0, 0, 0, 1, 1, 1, 0, 45, 0, 0, 214689, 0, 214689, 0, 0, 45, 0, 1

Offset: 0

Alois P. Heinz, Dec 09 2013

			A(5,2) = A(2,5) = 5:
  ._________. ._________. ._________. ._________. ._________.
  |_________| | ._____| | | |_____. | |   ._|   | |   |_.   |
  |_________| |_|_______| |_______|_| |___|_____| |_____|___|.
Square array A(n,k) begins:
  1, 1,  1,    1,      1,         1,          1, ...
  1, 0,  0,    0,      0,         1,          0, ...
  1, 0,  0,    0,      0,         5,          0, ...
  1, 0,  0,    0,      0,        56,          0, ...
  1, 0,  0,    0,      0,       501,          0, ...
  1, 1,  5,   56,    501,      4006,      27950, ...
  1, 0,  0,    0,      0,     27950,          0, ...
  1, 0,  0,    0,      0,    214689,          0, ...
  1, 0,  0,    0,      0,   1696781,          0, ...
  1, 0,  0,    0,      0,  13205354,          0, ...
  1, 1, 45, 7670, 890989, 101698212, 7845888732, ...
  ...

Liang Kai, Antidiagonals n = 0..26, flattened (Antidiagonals n = 0..17 from Alois P. Heinz)
R. S. Harris, Counting Polyomino Tilings
Liang Kai, Solving tiling enumeration problems by tensor network contractions, arXiv:2503.17698 [math.CO], 2025.
Wikipedia, Pentomino

Columns (or rows) include: A000012, A054318, A233428, A233429, A174249, A233430.
Cf. A099390, A233320, A230031, A246902, A247117, A278657.
Row sums of A247702, A247703, A247704, A247705, A247706, A247707, A247708, A247709, A247710, A247711, A247712, A247713 give A(n,5).

A(n,k) = 0 <=> n*k mod 5 > 0.

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.

Original entry on oeis.org

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

A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

Original entry on oeis.org

1, 1, 4, 23, 117, 454, 2003, 9157, 40899, 179399, 796558, 3546996, 15747348, 69834517, 310058192, 1376868145, 6112247118, 27132236455, 120453362938, 534754586459, 2373975139658, 10538953415410, 46786795734201, 207705902269424, 922089495910044, 4093525019450760

Offset: 0

Bob Harris (me13013(AT)gmail.com), Mar 13 2010

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
S. Butler, J. Ekstrand, and S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013.
R. S. Harris, Counting Nonomino Tilings and Other Things of that Ilk, G4G9 Gift Exchange book, 2010.
R. S. Harris, Counting Polyomino Tilings.
Wikipedia, Tetris.
Wikipedia, Tetromino.
Index entries for linear recurrences with constant coefficients, signature (2, 8, 8, 54, -77, -290, -76, -548, 469, 2258, 414, 1970, -1053, -6885, -1620, -3349, 1102, 9566, 3210, 2786, -489, -6047, -2600, -1102, 60, 1476, 659, 225, 39, -123, -50, -13, -3, 3, 1).

Cf. A134438, A174249, A226322, A232497, A232684, A232698, A232722, A233191, A233266.

Column k=4 of A230031.

G.f.: -(x^31 +3*x^30 -2*x^29 -7*x^28 -25*x^27 -78*x^26 +23*x^25 +116*x^24 +217*x^23 +604*x^22 -21*x^21 -556*x^20 -649*x^19 -1621*x^18 -175*x^17 +727*x^16 +523*x^15 +1707*x^14 +236*x^13 -470*x^12 -143*x^11 -749*x^10 -133*x^9 +166*x^8 +15*x^7 +126*x^6 +27*x^5 -23*x^4 -x^3 -6*x^2 -x +1) / (x^35 +3*x^34 -3*x^33 -13*x^32 -50*x^31 -123*x^30 +39*x^29 +225*x^28 +659*x^27 +1476*x^26 +60*x^25 -1102*x^24 -2600*x^23 -6047*x^22 -489*x^21 +2786*x^20 +3210*x^19 +9566*x^18 +1102*x^17 -3349*x^16 -1620*x^15 -6885*x^14 -1053*x^13 +1970*x^12 +414*x^11 +2258*x^10 +469*x^9 -548*x^8 -76*x^7 -290*x^6 -77*x^5 +54*x^4 +8*x^3 +8*x^2 +2*x -1). - Alois P. Heinz, Nov 26 2013

a(0) inserted, a(11)-a(22) from Alois P. Heinz, May 07 2013

a(23)-a(25) from Alois P. Heinz, Nov 26 2013

A233320 Number A(n,k) of tilings of a k X n rectangle using trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 3, 3, 0, 1, 1, 0, 0, 10, 0, 0, 1, 1, 1, 0, 23, 23, 0, 1, 1, 1, 0, 11, 62, 0, 62, 11, 0, 1, 1, 0, 0, 170, 0, 0, 170, 0, 0, 1, 1, 1, 0, 441, 939, 0, 939, 441, 0, 1, 1, 1, 0, 41, 1173, 0, 8342, 8342, 0, 1173, 41, 0, 1

Offset: 0

Alois P. Heinz, Dec 07 2013

Every row and column satisfies a linear recurrence. - Peter Kagey, Jul 17 2019

			Square array A(n,k) begins:
  1, 1,  1,    1,   1,    1,       1, ...
  1, 0,  0,    1,   0,    0,       1, ...
  1, 0,  0,    3,   0,    0,      11, ...
  1, 1,  3,   10,  23,   62,     170, ...
  1, 0,  0,   23,   0,    0,     939, ...
  1, 0,  0,   62,   0,    0,    8342, ...
  1, 1, 11,  170, 939, 8342,   80092, ...
  1, 0,  0,  441,   0,    0,  614581, ...
  1, 0,  0, 1173,   0,    0, 5271923, ...

Liang Kai, Antidiagonals n = 0..35, flattened (Antidiagonals 0..28 from Alois P. Heinz)
Wikipedia, Tromino

Columns (or rows) include: A000012, A001835, A134438, A233339, A233340, A233290, A233343, A269958, A215826, A269959, A269664.

Cf. A099390, A230031, A233427, A270061, A364457.

A(n,k) = 0 <=> n*k mod 3 > 0.

A233191 Number of tilings of a 4 X n rectangle using L and T tetrominoes.

Original entry on oeis.org

1, 0, 2, 4, 12, 16, 76, 128, 386, 832, 2368, 5024, 13946, 31680, 82632, 193696, 498174, 1182464, 2993384, 7213648, 18061074, 43832960, 109163384, 266217472, 660116398, 1615451648, 3995295112, 9796774896, 24189684402, 59396496000, 146494223160, 360026507808

Offset: 0

Alois P. Heinz, Dec 05 2013

			a(3) = 4:
._____.  ._____.  ._____.  ._____.
|_. ._|  |_. ._|  | |_. |  | ._| |
| |_| |  | |_| |  | ._| |  | |_. |
| ._| |  | |_. |  |_| |_|  |_| |_|
|_|___|  |___|_|  |_____|  |_____|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (0, 4, 4, 5, -8, 4, 12, -18, -8, 0, 0, -8).

Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233266.

gf:= (2*x^6+x^4+2*x^2-1) / (-8*x^12 -8*x^9 -18*x^8
     +12*x^7 +4*x^6 -8*x^5 +5*x^4 +4*x^3 +4*x^2 -1):
a:= n-> coeff(series(gf, x, n+1), x, n):
seq(a(n), n=0..40);

G.f.: (2*x^6+x^4+2*x^2-1) / (-8*x^12 -8*x^9 -18*x^8 +12*x^7 +4*x^6 -8*x^5 +5*x^4 +4*x^3 +4*x^2 -1).

A233266 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, T, Z.

Original entry on oeis.org

1, 0, 2, 10, 24, 70, 276, 820, 2616, 8702, 27902, 89500, 291050, 939222, 3029950, 9798606, 31657182, 102237766, 330356240, 1067310022, 3447911968, 11139391996, 35988377472, 116265759012, 375619824338, 1213515477460, 3920484872552, 12665878390278

Offset: 0

Alois P. Heinz, Dec 06 2013

			a(3) = 10:
._____.  ._____.  ._____.  ._____.  ._____.
| |_. |  | ._| |  | .___|  |___. |  | .___|
|_. | |  | | ._|  |_| | |  | | |_|  |_| ._|
| |_|_|  |_|_| |  | ._| |  | |_. |  |___| |
|_____|  |_____|  |_|___|  |___|_|  |_____|
._____.  ._____.  ._____.  ._____.  ._____.
| ._| |  | |_. |  |_. ._|  |_. ._|  |___. |
| |_. |  | ._| |  | |_| |  | |_| |  |_. |_|
|_| |_|  |_| |_|  | |_. |  | ._| |  | |___|
|_____|  |_____|  |___|_|  |_|___|  |_____|.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 10.
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (2, 4, 4, -6, -28, 15, 4, -13, 20, -4, 16, -10, 8, -2).

Cf. A084480, A174248, A226322, A230031, A232497, A233139, A233191, A242636.

G.f.: (x^8 -4*x^7 +3*x^6 -2*x^5 -2*x^4 -2*x^3 +2*x^2 +2*x -1) / (-2*x^14 +8*x^13 -10*x^12 +16*x^11 -4*x^10 +20*x^9 -13*x^8 +4*x^7 +15*x^6 -28*x^5 -6*x^4 +4*x^3 +4*x^2 +2*x -1).

A360499 Number of ways to tile an n X n square using rectangles with distinct dimensions.

Original entry on oeis.org

1, 1, 21, 269, 4489, 82981, 2995185, 118897973

Offset: 1

Scott R. Shannon, Feb 09 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

			a(1) = 1 as the only way to tile a 1 x 1 square is with a square with dimensions 1 x 1.
a(2) = 1 as the only way to tile a 2 x 2 square is with a square with dimensions 2 x 2.
a(3) = 21. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |   |       |   |       |   |   |           |
  +           +   +---+---+---+   +---+---+   +   +---+---+---+
  |           |   |           |   |       |   |   |           |
  +           +   +           +   +       +   +   +           +
  |           |   |           |   |       |   |   |           |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 1 way, the second in 8 different ways, the third in 8 different ways and the fourth in 4 different ways, giving 21 ways in total.

Cf. A360498 (oblongs), A182275 (not necessarily distinct dimensions), A004003, A099390, A065072, A233320, A230031.

A232684 Number of tilings of a 6 X 2n rectangle with 3n tetrominoes of any shape.

Original entry on oeis.org

1, 9, 2003, 178939, 22483347, 2569437089, 304446920314, 35704534261665, 4203065267122878, 494232382069456694, 58138539945306221167, 6838279451118114249916, 804352962762109905924360, 94610929453211737452277488, 11128526714790919845521179844

Offset: 0

Alois P. Heinz, Nov 27 2013

nonn

			a(1) = 9:
.___. .___. .___. .___. .___. .___. .___. .___. .___.
|   | | | | |   | |_. | |   | | ._| |   | | ._| |_. |
|___| | | | |___| | | | |___| | | | |___| | | | | | |
|   | | | | | | | | |_| |_. | |_| | | ._| |_| | | |_|
|___| |_|_| | | | |___| | | | |___| | | | | | | | | |
|   | |   | | | | |   | | |_| |   | |_| | | |_| |_| |
|___| |___| |_|_| |___| |___| |___| |___| |___| |___|.

Alois P. Heinz, Table of n, a(n) for n = 0..300
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
Wikipedia, Tetromino

Cf. A174248, A232698, A232722.

Even bisection of column k=6 of A230031.

A232698 Number of tilings of an 8 X n rectangle with 2n tetrominoes of any shape.

Original entry on oeis.org

1, 1, 25, 997, 40899, 800290, 22483347, 657253434, 19077209438, 517312744806, 14571957312254, 412240433359025, 11632857444709188, 326275845576101452, 9187549952207915190, 258821654387452112268, 7288072624408347082481, 205113474464891986564786

Offset: 0

Alois P. Heinz, Nov 27 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300
S. Butler, J. Ekstrand, S. Osborne, TETRIS Tiling, AMS Spring Central Sectional, Iowa State University, April 27-28 2013
Wikipedia, Tetromino

Cf. A174248, A232684, A232722.

Column k=8 of A230031.

cp's OEIS Frontend

A007598 Squared Fibonacci numbers: a(n) = F(n)^2 where F = A000045.

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A233427 Number A(n,k) of tilings of a k X n rectangle using pentominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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A174248 Number of tilings of a 4 X n rectangle with n tetrominoes of any shape.

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A233320 Number A(n,k) of tilings of a k X n rectangle using trominoes of any shape; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A233191 Number of tilings of a 4 X n rectangle using L and T tetrominoes.

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A233266 Number of tilings of a 4 X n rectangle using tetrominoes of shapes L, T, Z.

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