A127865 -id:A127865 - cp's OEIS frontend

A127864 Number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

1, 1, 5, 11, 33, 87, 241, 655, 1793, 4895, 13377, 36543, 99841, 272767, 745217, 2035967, 5562369, 15196671, 41518081, 113429503, 309895169, 846649343, 2313089025, 6319476735, 17265131521, 47169216511, 128868696065, 352075825151, 961889042433, 2627929735167

Offset: 0

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

The signed version of this sequence appears as A077917.

			a(2) = 5 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
S. Heubach, Tiling with Ls and Squares, 2005.
Index entries for linear recurrences with constant coefficients, signature (1,4,2).

Cf. A077917, A127865, A127866, A127867, A127868, A127869, A127870, A127871, A127872.

Column k=2 of A220054. - Alois P. Heinz, Dec 03 2012

I:=[1,1,5]; [n le 3 select I[n] else Self(n-1) + 4*Self(n-2) + 2*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2022

CoefficientList[Series[1/(1-x-4*x^2-2*x^3), {x,0,30}], x]

A028860 = BinaryRecurrenceSequence(2,2,-1,1)
def A127864(n): return A028860(n+2) + (-1)^n
[A127864(n) for n in range(51)] # G. C. Greubel, Dec 08 2022

a(n) = a(n-1) + 4*a(n-2) + 2*a(n-3).
a(n) = (-1)^n + (1/sqrt(3)) * ((1+sqrt(3))^n - (1-sqrt(3))^n).
G.f.: 1/(1 - x - 4*x^2 - 2*x^3).
a(n) = A028860(n+2) + (-1)^n. - R. J. Mathar, Oct 29 2010
E.g.f.: exp(-x) + (2/sqrt(3))*exp(x)*sinh(sqrt(3)*x). - G. C. Greubel, Dec 08 2022
From Greg Dresden, Nov 10 2024: (Start)
a(n) = 1 + 4*a(n-2) + 6*Sum_{i=0..n-3} a(i) for n>1.
a(2*n) = a(n)^2 + 4*a(n-1)^2 + 4*a(n-1)*a(n-2) for n>1. (End)

A127867 Number of tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

1, 1, 11, 39, 195, 849, 3895, 17511, 79339, 358397, 1620843, 7326991, 33127155, 149766353, 677103839, 3061202815, 13839823275, 62570318397, 282882722979, 1278922980071, 5782057329219, 26140890761969, 118183916056327, 534313772133687, 2415651952691819

Offset: 0

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

nonn

			a(2) = 11 because the 3 X 2 board can be tiled in one way with only square tiles, in 8 ways using one L-tile and 3 square tiles and in 2 ways with 2 L-tiles.

Alois P. Heinz, Table of n, a(n) for n = 0..500
P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
Index entries for linear recurrences with constant coefficients, signature (3, 7, -1, 2).

Cf. A127864, A127865, A127866, A127868, A127869, A127870.

Column k=3 of A220054. - Alois P. Heinz, Dec 03 2012

Table[Coefficient[Normal[Series[(1 - x)^2/(1 - 3x - 7x^2 + x^3 - 2x^4), {x, 0, 30}]], x, n], {n, 0, 30}]

G.f.: (1-x)^2/(1-3x-7x^2+x^3-2x^4).

A127870 Number of tilings of a 4 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

1, 1, 33, 195, 2023, 16839, 151817, 1328849, 11758369, 103628653, 914646205, 8068452381, 71189251649, 628067760289, 5541284098945, 48888866203241, 431331449340441, 3805499681885145, 33574725778806817, 296219181642118401, 2613448287490035073

Offset: 0

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

nonn

			a(2) = 33 because the 4x2 board can be tiled in one way with only square tiles, in 12 ways using one L-tile and 5 square tiles and in 20 ways with 2 L-tiles and 2 square tiles.

Alois P. Heinz, Table of n, a(n) for n = 0..500
P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
Index entries for linear recurrences with constant coefficients, signature (5, 34, 6, -72, -28, 74, -10, -4, -4).

Cf. A127864, A127865, A127866, A127867, A127868, A127869.

Column k=4 of A220054. - Alois P. Heinz, Dec 03 2012

Table[Coefficient[Normal[Series[(1 - 4 z - 6 z^2 - 10 z^3 - 8 z^4 - 4 z^5)/(1 - 5z - 34 z^2 - 6 z^3 + 72 z^4 + 28 z^5 - 74 z^6 + 10 z^7 + 4 z^8 + 4 z^9), {x, 0, 30}]], x, n], {n, 0, 30}]

G.f.: (1 - 4 z - 6 z^2 - 10 z^3 - 8 z^4 - 4 z^5) / (1 - 5z - 34 z^2 - 6 z^3 + 72 z^4 + 28 z^5 - 74 z^6 + 10 z^7 + 4 z^8 + 4 z^9).

A127866 Number of L-shaped tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

4, 12, 52, 172, 580, 1852, 5828, 17980, 54788, 165116, 493316, 1463036, 4312068, 12641276, 36887556, 107201532, 310427652, 896045052, 2579017732, 7403843580, 21205303300, 60604891132, 172872744964, 492233179132, 1399272374276

Offset: 2

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

			a(2) = 4 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile and thus all the tilings of the 2 X 2 board contain 4 L-shaped tiles.

P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
Index entries for linear recurrences with constant coefficients, signature (3, 4, -8, -12, -4).

Cf. A127864, A127865, A127867, A127868, A127869, A127870, A127871, A127872.

Table[Coefficient[Normal[Series[4x^2/((1 + x)(1 - 2x - 2x^2)^2), {x, 0, 20}]], x, n], {n, 0, 20}]

a(n) = 4 (-1)^n - (2/9)[(9-5*Sqrt(3))(1+Sqrt(3))^n + (9+5*Sqrt(3))(1-Sqrt(3))^n] - (n/3)[(1-Sqrt(3))(1+Sqrt(3))^n+ (1+Sqrt(3))(1-Sqrt(3))^n].

G.f.: 4x^2/((1+x)(1-2x-2x^2)^2).

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

A127868 Number of square tiles in all tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

3, 30, 171, 1044, 5691, 30678, 159891, 821100, 4151511, 20764590, 102880755, 505866804, 2471159019, 12004723878, 58037429739, 279405305676, 1340130574407, 6406579480446, 30536794325547, 145166910196116, 688444702671291, 3257788855054518, 15385512460164963

Offset: 1

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

nonn

			a(2) = 30 because the 3 X 2 board can be tiled in one way with only square tiles, in 8 ways using one L-tile and 3 square tiles and in 2 ways with 2 L-tiles, so there are altogether 6 + 8*3 = 30 square tiles in all of the 3x2 tilings.

Robert Israel, Table of n, a(n) for n = 1..1510
P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
S. Heubach, Tiling with Ls and Squares, 2005.
Index entries for linear recurrences with constant coefficients, signature (6,5,-44,-39,2,-29,4,-4).

Cf. A127864, A127865, A127866, A127867, A127869, A127870.

I:=[3,30,171,1044,5691,30678,159891,821100]; [n le 8 select I[n] else 6*Self(n-1)+5*Self(n-2)-44*Self(n-3)-39*Self(n-4)+2*Self(n-5)-29*Self(n-6)+4*Self(n-7)-4*Self(n-8): n in [1..30]]; // Vincenzo Librandi, Dec 23 2015

f:= gfun:-rectoproc({a(n) - 6*a(n-1)-5*a(n-2)+44*a(n-3)+39*a(n-4)-2*a(n-5)+29*a(n-6)-4*a(n-7)+4*a(n-8), a(0) = 0, a(1) = 3, a(2) = 30, a(3) = 171, a(4) = 1044, a(5) = 5691, a(6) = 30678, a(7) = 159891},a(n), remember):
seq(f(n), n=1..40); # Robert Israel, Dec 22 2015

Table[Coefficient[Normal[Series[3x(1-x)^2(1+6x+3x^2)/(1-3x-7x^2+x^3-2x^4)^2, {x, 0, 30}]], x, n], {n, 0, 30}]
LinearRecurrence[{6, 5, -44, -39, 2, -29, 4, -4}, {3, 30, 171, 1044, 5691, 30678, 159891, 821100}, 25] (* Vincenzo Librandi, Dec 23 2015 *)

my(x='x+O('x^100)); Vec(3*x*(1-x)^2*(1+6*x+3*x^2)/(1-3*x-7*x^2+x^3-2*x^4)^2) \\ Altug Alkan, Dec 22 2015

G.f.: 3x(1-x)^2(1+6x+3x^2)/(1-3x-7x^2+x^3-2x^4)^2.

A127869 Number of L-shaped tiles in all tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

Original entry on oeis.org

12, 60, 432, 2348, 13144, 69280, 361012, 1841736, 9286900, 46303316, 228903592, 1123242916, 5477879120, 26572232312, 128302070508, 616985221280, 2956362520140, 14120605179500, 67252176519008, 319477138444252, 1514116534887688, 7160712605686480, 33799490762646948

Offset: 2

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

nonn

			a(2) = 12 because the 3 X 2 board can be tiled in one way with only square tiles, in 8 ways using one L-tile and 3 square tiles and in 2 ways with 2 L-tiles, so there are altogether 8 + 2*2 = 12 L-tiles in all of the 3 X 2 tilings.

P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
S. Heubach, Tiling with Ls and Squares, 2005.
Index entries for linear recurrences with constant coefficients, signature (6,5,-44,-39,2,-29,4,-4).

Cf. A127864, A127865, A127866, A127867, A127868, A127870.

Table[Coefficient[Normal[Series[4x^2(3-3x+3x^2-4x^3+x^4)/(1-3x-7x^2+x^3-2x^4)^2, {x, 0, 30}]], x, n], {n, 0, 30}]

G.f.: 4*x^2*(x-1)*(x^3-3*x^2-3)/(1-3*x-7*x^2+x^3-2*x^4)^2.

A245965 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k 1 X 1 tiles.

Original entry on oeis.org

1, 0, 0, 1, 0, 4, 0, 0, 1, 2, 0, 0, 8, 0, 0, 1, 0, 0, 20, 0, 0, 12, 0, 0, 1, 0, 16, 0, 0, 54, 0, 0, 16, 0, 0, 1, 4, 0, 0, 112, 0, 0, 104, 0, 0, 20, 0, 0, 1, 0, 0, 108, 0, 0, 352, 0, 0, 170, 0, 0, 24, 0, 0, 1, 0, 48, 0, 0, 664, 0, 0, 800, 0, 0, 252, 0, 0, 28, 0, 0, 1, 8, 0, 0, 704, 0, 0, 2280, 0, 0, 1520, 0, 0, 350, 0, 0, 32, 0, 0, 1, 0, 0, 416, 0, 0, 4064, 0, 0, 5820, 0, 0, 2576, 0, 0, 464, 0, 0, 36, 0, 0, 1

Offset: 0

Emeric Deutsch, Aug 15 2014

Row n has 2n+1 entries.
Sum of entries in row n = A127864(n).
Sum_{k>=0} k*T(n,k) = A127865(n).

			T(2,1)=4 because we can place the 1 X 1 tile in any corner of the 2 X 2 board.
Triangle starts:
  1;
  0,  0,  1;
  0,  4,  0,  0,  1;
  2,  0,  0,  8,  0,  0,  1;
  0,  0, 20,  0,  0, 12,  0,  0,  1;

P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8

Cf. A127864, A127865, A245966.

G := 1/(1-t^2*z-4*t*z^2-2*z^3): Gser := simplify(series(G, z = 0, 15)): for j from 0 to 13 do P[j] := sort(coeff(Gser, z, j)) end do: for j from 0 to 13 do seq(coeff(P[j], t, i), i = 0 .. 2*j) end do; # yields sequence in triangular form

G.f.: 1/(1-t^2*z - 4*t*z^2 - 2*z^3).
The trivariate g.f. with z marking length, t marking 1 X 1 tiles, and s marking L-shaped tiles is 1/(1-t^2*z-4*t*s*z^2-2*s^2*z^3).
From Robert Israel, Aug 15 2014: (Start)
T(n+3,k+2) = T(n+2,k) + 4*T(n+1,k+1) + 2*T(n,k+2).
T(n,0) = 2^(n/3) if n == 0 (mod 3), T(n,0) = 0 otherwise.
T(n,1) = (n+1)*2^((n+4)/3)/3 if n == 2 (mod 3), T(n,1) = 0 otherwise.
(End)

A245966 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k L-shaped tiles.

Original entry on oeis.org

1, 1, 1, 4, 1, 8, 2, 1, 12, 20, 1, 16, 54, 16, 1, 20, 104, 112, 4, 1, 24, 170, 352, 108, 1, 28, 252, 800, 664, 48, 1, 32, 350, 1520, 2280, 704, 8, 1, 36, 464, 2576, 5820, 4064, 416, 1, 40, 594, 4032, 12404, 14784, 4560, 128, 1, 44, 740, 5952, 23408, 41104, 25376, 3200, 16

Offset: 0

Emeric Deutsch, Aug 15 2014

Row n contains 1+floor(2n/3) entries.
Sum of entries in row n = A127864(n).
Sum_{k>=0} k*T(n,k) = A127866(n).

			T(2,1) = 4 because we can place the L-shaped tile in the 2*2 board in 4 positions.
Triangle starts:
  1;
  1;
  1,  4;
  1,  8,  2;
  1, 12, 20;
  1, 16, 54, 16;

P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8

Cf. A127864, A127865, A245965.

G := 1/(1-z-4*t*z^2-2*t^2*z^3): Gser := simplify(series(G, z = 0, 15)): for j from 0 to 13 do P[j] := sort(coeff(Gser, z, j)) end do: for j from 0 to 13 do seq(coeff(P[j], t, i), i = 0 .. floor(2*j*(1/3))) end do; # yields sequence in triangular form

G.f.: 1/(1 - z - 4*t*z^2 - 2*t^2*z^3).

The trivariate g.f. with z marking length, t marking 1 X 1 tiles, and s marking L-shaped tiles is 1/(1 - t^2*z - 4*t*s*z^2 - 2*s^2*z^3).

cp's OEIS Frontend

A127864 Number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127867 Number of tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127870 Number of tilings of a 4 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127866 Number of L-shaped tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127868 Number of square tiles in all tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A127869 Number of L-shaped tiles in all tilings of a 3 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).

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A245965 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k 1 X 1 tiles.

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A245966 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k L-shaped tiles.