A054854 -id:A054854 - cp's OEIS frontend

A190759 Number of tilings of a 5 X n rectangle using right trominoes and 2 X 2 tiles.

1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120, 6912, 60904, 75136, 491960, 720640, 4023592, 6828928, 32819320, 63472640, 270471784, 574543744, 2256221368, 5119155712, 18940876712, 45266369152, 159625747960, 397949457408, 1350573713256

Offset: 0

Alois P. Heinz, May 18 2011

			a(2) = 4, because there are 4 tilings of a 5 X 2 rectangle using right trominoes and 2 X 2 tiles:
.___. .___. .___. .___.
| . | | . | | ._| |_. |
|___| |___| |_| | | |_|
| ._| |_. | |___| |___|
|_| | | |_| | . | | . |
|___| |___| |___| |___|

Alois P. Heinz, Table of n, a(n) for n = 0..650
Index entries for linear recurrences with constant coefficients, signature (0, 13, 2, -57, -12, 190, -10, -453, 396, -2, -88, 308, -160, -80).

Cf. A165799, A165791, A165716, A054854, A054856.

Column k=5 of A219946.

a:= n-> (Matrix(14, (i, j)-> `if`(i=j-1, 1, `if`(i=14, [-80, -160, 308, -88, -2, 396, -453, -10, 190, -12, -57, 2, 13, 0][j], 0)))^n. <<0, 1/4, 0, 1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120>>)[4,1]: seq(a(n), n=0..30);

a[n_] := (MatrixPower[ Table[ If[i == j-1, 1, If[i == 14, {-80, -160, 308, -88, -2, 396, -453, -10, 190, -12, -57, 2, 13, 0}[[j]], 0]], {i, 1, 14}, {j, 1, 14}], n] . {0, 1/4, 0, 1, 0, 4, 0, 16, 0, 136, 0, 1128, 384, 8120})[[4]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 05 2013, translated from Alois P. Heinz's Maple program *)

G.f.: (20*x^12+40*x^11 +18*x^10+52*x^9 +35*x^8-26*x^7 +34*x^6-4*x^5 -21*x^4 +2*x^3 +9*x^2-1) / (-80*x^14-160*x^13 +308*x^12-88*x^11 -2*x^10+396*x^9 -453*x^8-10*x^7 +190*x^6-12*x^5 -57*x^4+2*x^3 +13*x^2-1).

A046672 Expansion of 1/(1-2x-3x^2+2*x^3).

Original entry on oeis.org

1, 2, 7, 18, 53, 146, 415, 1162, 3277, 9210, 25927, 72930, 205221, 577378, 1624559, 4570810, 12860541, 36184394, 101808791, 286449682, 805956949, 2267645362, 6380262207, 17951546602, 50508589101, 142111293594, 399845261287, 1125007225154

Offset: 0

N. J. A. Sloane, Nov 17 2002

Iain Fox, Table of n, a(n) for n = 0..2226
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 6.
Index entries for linear recurrences with constant coefficients, signature (2, 3, -2).

Partial sums of A054854.

Cf. A101197.

first(n) = Vec(1/(1-2*x-3*x^2+2*x^3) + O(x^n)) \\ Iain Fox, Dec 02 2017

a(n) = 2*a(n-1) + 3*a(n-2) - 2*a(n-3), n > 2. - Iain Fox, Dec 02 2017

A063651 Number of ways to tile a 7 X n rectangle with 1 X 1 and 2 X 2 tiles.

Original entry on oeis.org

1, 1, 21, 85, 747, 4375, 31387, 202841, 1382259, 9167119, 61643709, 411595537, 2758179839, 18448963469, 123518353059, 826573277157, 5532716266089, 37028886137273, 247839719105625, 1658772577825883, 11102227136885119

Offset: 0

Reiner Martin, Jul 23 2001

Index entries for linear recurrences with constant coefficients, signature (3,30,-17,-138,85,116,-42,-32).

Cf. A001045, A054854, A054855, A063650-A063654.

Column k=7 of A245013.

a(n) = 3a(n-1) + 30a(n-2) - 17a(n-3) - 138a(n-4) + 85a(n-5) + 116a(n-6) - 42a(n-7) - 32a(n-8). - Keith Schneider (kschneid(AT)bulldog.unca.edu), Apr 02 2006

G.f.: ( 1 -2*x -12*x^2 +9*x^3 +17*x^4 -6*x^5 -6*x^6 ) / ( 1 -3*x -30*x^2 +17*x^3 +138*x^4 -85*x^5 -116*x^6 +42*x^7 +32*x^8 ). - Colin Barker, Nov 29 2012

A128101 Triangle read by rows: T(n,k) is the number of ways to tile a 4 X n rectangle with k pieces of 2 X 2 tiles and 4(n-k) pieces of 1 X 1 tiles (0<=k<=2*floor(n/2)).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 6, 4, 1, 9, 16, 8, 1, 1, 12, 37, 34, 9, 1, 15, 67, 105, 65, 15, 1, 1, 18, 106, 248, 250, 108, 16, 1, 21, 154, 490, 726, 522, 176, 24, 1, 1, 24, 211, 858, 1736, 1824, 994, 260, 25, 1, 27, 277, 1379, 3604, 5148, 4090, 1770, 385, 35, 1, 1, 30, 352, 2080

Offset: 0

Emeric Deutsch, Feb 19 2007

Row 2n has 2n+1 terms; row 2n+1 has 2n+1 terms.

			Triangle starts:
1;
1;
1,3,1;
1,6,4;
1,9,16,8,1;
1,12,37,34,9;
1,15,67,105,65,15,1;

S. Heubach, Tiling an m X n area with squares of size up to k X k (m <= 5), Congressus Numerantium 140 (1999), pp. 43-64.

R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares arXiv:1609.03964 (2016)

Cf. A054854 (row sums), A008585, A080855, A128102.

G:=(1-t*z)/(1-z-t*z-2*t*z^2-t^2*z^2+t^2*z^3+t^3*z^3): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..2*floor(n/2)) od; # yields sequence in triangular form

CoefficientList[#, t]& /@ CoefficientList[(1 - t z)/(1 - z - t z - 2 t z^2 - t^2 z^2 + t^2 z^3 + t^3 z^3) + O[z]^12, z]  // Flatten (* Jean-François Alcover, Aug 07 2018 *)

G.f.=(1-tz)/(1-z-tz-2tz^2-t^2*z^2+t^2*z^3+t^3*z^3).
Sum (T(n,k), k=0..2*floor(n/2) ) = A054854(n).
T(n,1)=3(n-1)=A008585(n-1).
T(n,2)=A080855(n-2).
Sum(k*T(n,k), k=0..2*floor(n/2)) = A128102(n).
T(n,3) = (n-3)*(9*n^2-63*n+124)/2, n>=3. - R. J. Mathar, Aug 23 2016
T(n,4) = (3*n-13)*(9*n^3-123*n^2+602*n-1024)/8, n>=4. - R. J. Mathar, Aug 23 2016

A179618 T(n,k) = Half the number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock diagonal sum differing from its antidiagonal sum by more than 2.

Original entry on oeis.org

5, 11, 11, 21, 35, 21, 43, 93, 93, 43, 85, 269, 314, 269, 85, 171, 747, 1213, 1213, 747, 171, 341, 2115, 4375, 6427, 4375, 2115, 341, 683, 5933, 16334, 31387, 31387, 16334, 5933, 683, 1365, 16717, 59925, 159651, 202841, 159651, 59925, 16717, 1365, 2731

Offset: 1

R. H. Hardin, Jan 10 2011

T(n,k) apparently is also the number of ways to tile an (n+2) X (k+2) rectangle with 1 X 1 and 2 X 2 tiles.

			Table starts
     5     11      21        43         85         171           341
    11     35      93       269        747        2115          5933
    21     93     314      1213       4375       16334         59925
    43    269    1213      6427      31387      159651        795611
    85    747    4375     31387     202841     1382259       9167119
   171   2115   16334    159651    1382259    12727570     113555791
   341   5933   59925    795611    9167119   113555791    1355115601
   683  16717  221799   4005785   61643709  1029574631   16484061769
  1365  47003  817280  20064827  411595537  9258357134  198549329897
  2731 132291 3018301 100764343 2758179839 83605623809 2403674442213
Some solutions for 6 X 6:
  0 2 0 2 0 2    0 1 0 2 1 2    0 2 0 2 0 2    0 1 0 2 0 1
  2 0 2 0 2 1    2 0 2 0 2 0    2 0 1 0 1 0    2 0 2 0 2 0
  0 2 0 2 0 2    1 2 1 2 0 2    0 2 0 2 0 2    0 2 0 2 0 2
  2 0 2 0 2 1    2 0 2 0 1 0    1 0 2 0 2 0    1 0 2 0 2 0
  0 2 0 2 0 2    0 2 0 2 0 2    0 2 0 2 0 2    0 2 1 2 1 2
  1 0 1 0 1 0    2 1 2 1 2 0    2 1 2 1 2 1    2 0 2 0 2 0

R. H. Hardin, Table of n, a(n) for n = 1..839

Diagonal is A063443(n+2).
Column 1 is A001045(n+3).
Column 2 is A054854(n+2).
Column 3 is A054855(n+2).
Column 4 is A063650(n+2).
Column 5 is A063651(n+2).
Column 6 is A063652(n+2).
Column 7 is A063653(n+2).
Column 8 is A063654(n+2).

cp's OEIS Frontend

A190759 Number of tilings of a 5 X n rectangle using right trominoes and 2 X 2 tiles.

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A046672 Expansion of 1/(1-2*x-3*x^2+2*x^3).

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A063651 Number of ways to tile a 7 X n rectangle with 1 X 1 and 2 X 2 tiles.

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A128101 Triangle read by rows: T(n,k) is the number of ways to tile a 4 X n rectangle with k pieces of 2 X 2 tiles and 4(n-k) pieces of 1 X 1 tiles (0<=k<=2*floor(n/2)).

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A179618 T(n,k) = Half the number of (n+1) X (k+1) 0..2 arrays with every 2 X 2 subblock diagonal sum differing from its antidiagonal sum by more than 2.

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A046672 Expansion of 1/(1-2x-3x^2+2*x^3).