A063443 -id:A063443 - cp's OEIS frontend

A140314 Number of n X n binary matrices containing no more than two 1's in any 2 X 2 subblock.

1, 2, 11, 195, 8969, 1232795, 471487297, 516844071813, 1603508113421239, 14148460147460008963, 354360105981877906958557, 25210728065407321182797745205, 5093486498660018140333056634922039

Offset: 0

R. H. Hardin, May 25 2008

nonn

No more than 1 1 in any 2 X 2 subblock gives A063443.

No more than 3 1's in any 2 X 2 subblock gives A139810.

R. H. Hardin, May 25 2008, Table of n, a(n) for n = 0..15

Cf. A063443, A139810.

A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

Original entry on oeis.org

2, 16, 288, 11664, 1458000, 506250000, 414720000000, 869730877440000, 5045702916833280000, 77297454895962562560000, 3017525202366485003182080000, 307389127582207654481154908160000, 83016370640108703579427655610531840000, 58770343311359208383258439665073059266560000

Offset: 1

Vaclav Kotesovec, May 05 2012

nonn

Semi-knight is a semi-leaper [1,2]. Moves of a semi-knight are allowed only in [2,1] and [-2,-1]. See also semi-bishops (A187235).

Vaclav Kotesovec, Table of n, a(n) for n = 1..60
V. Kotesovec, Non-attacking chess pieces

Cf. A067962, A067966, A063443, A006506, A067965, A066864, A067963, A067964, A182563

Table[If[EvenQ[n],Fibonacci[n/2+2]^(n+2)*Product[Fibonacci[j+2]^4,{j,1,n/2-1}],Fibonacci[(n+1)/2+2]^((n+1)/2)*Fibonacci[(n-1)/2+2]^((n-1)/2)*Product[Fibonacci[j+2]^4,{j,1,(n-1)/2}]],{n,1,20}]

a(n) = F(n/2+2)^(n+2)*prod(j=1,n/2-1,F(j+2)^4) if n is even, F((n+1)/2+2)^((n+1)/2)*F((n-1)/2+2)^((n-1)/2)*prod(j=1,(n-1)/2,F(j+2)^4) if n is odd, where F(n) = A000045(n) is the n-th Fibonacci number.

a(n) is asymptotic to C^4*((1+sqrt(5))/2)^((n+2)*(n+4))/5^(3/2*(n+2)), where C=1.226742010720353244... is Fibonacci Factorial Constant, see A062073.

A275869 Number of nonequivalent ways to place k>=0 nonattacking kings on an n X n board.

Original entry on oeis.org

2, 2, 11, 51, 922, 25618, 1597350, 169413040, 33716225195, 11838480390673, 7588965091313449, 8705702554970941096, 18079208010076976255573, 67519585404524909086260614, 455193583190737164106702088892, 5527752160260327852724215089473548

Offset: 1

Heinrich Ludwig, Dec 11 2016

nonn

Also number of nonequivalent ways to tile an n+1 X n+1 square with 1 X 1 and 2 X 2 tiles.
Also row sum of triangle A236679.
Rotations and reflections of a placement are not counted. If they are to be counted, see A063443.

Andrew Howroyd, Table of n, a(n) for n = 1..30

Cf. A236679, A063443.

a(10)-a(16) from Andrew Howroyd, May 30 2017

A335560 Number of ways to tile an n X n square with 1 X 1 squares and (n-1) X 1 vertical or horizontal strips.

Original entry on oeis.org

1, 16, 131, 335, 851, 2207, 5891, 16175, 45491, 130367, 378851, 1112015, 3286931, 9762527, 29091011, 86879855, 259853171, 777986687, 2330814371, 6986151695, 20945872211, 62812450847, 188387020931, 565060399535, 1694979872051, 5084536963007, 15252805582691

Offset: 1

Oluwatobi Jemima Alabi, Jun 14 2020

It is assumed that 1 X 1 squares and 1 X 1 strips can be distinguished. - Alois P. Heinz, Feb 23 2022

			Here is one of the 131 ways to tile a 3 X 3 square, in this case using two horizontal and two vertical strips:
   _ _ _
  |_ _| |
  | |_|_|
  |_|_ _|

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A063443 and A211348 (tiling an n X n square with smaller squares).

Cf. A028420 (tiling an n X n square with monomers and dimers).

Join[{1, 16}, LinearRecurrence[{6, -11, 6}, {131, 335, 851}, 25]] (* Amiram Eldar, Jun 16 2020 *)

Vec(x*(1 + 10*x + 46*x^2 - 281*x^3 + 186*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Jun 14 2020

a(n) = 2*3^n + 12*2^n - 19, for n >= 3.
From Colin Barker, Jun 14 2020: (Start)
G.f.: x*(1 + 10*x + 46*x^2 - 281*x^3 + 186*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)).
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>5. (End)
E.g.f.: 5 - 19*exp(x) +12 *exp(2*x) + 2*exp(3*x) - 10*x - 31*x^2/2. - Stefano Spezia, Aug 25 2025

A353934 Number of tilings of an n X n square using right trominoes, dominoes, and monominoes.

Original entry on oeis.org

1, 1, 11, 369, 83374, 90916452, 546063639624, 17259079054003609, 2916019543694306398589, 2620143594924539083433405392, 12541344781693990981151732534871036, 319608708168951734031266758322647453517098, 43373075269161087186367095378869660507262626652634

Offset: 0

Alois P. Heinz, May 11 2022

nonn

			a(2) = 11:
  .___. .___. .___. .___. .___. .___. .___. .___. .___. .___. .___.
  |_|_| |___| | | | |_|_| |___| |_| | | |_| |_| | |_. | | ._| | |_|
  |_|_| |___| |_|_| |___| |_|_| |_|_| |_|_| |___| |_|_| |_|_| |___| .

Alois P. Heinz, Table of n, a(n) for n = 0..16
Wikipedia, Polyomino

Main diagonal of A353877.

Cf. A004003, A028420, A063443, A219874, A219952, A219994, A220061, A220778, A233807, A270071, A353777.

a(n) = A353877(n,n).

A141483 Number of n X n binary matrices, symmetric under horizontal and vertical reflection, with no more than 1 one in any 2 X 2 subblock.

Original entry on oeis.org

1, 2, 1, 5, 2, 35, 5, 314, 35, 6427, 314, 202841, 6427, 12727570, 202841, 1355115601, 12727570, 269718819131, 1355115601, 94707789944544, 269718819131, 60711713670028729, 94707789944544, 69645620389200894313

Offset: 0

R. H. Hardin, Aug 09 2008

nonn

a(2n-1) = A063443(n+1). - Vaclav Kotesovec, May 01 2012

a(2n) = a(2n-3), for n>1. - Vaclav Kotesovec, Apr 01 2016

R. H. Hardin and Vaclav Kotesovec, Table of n, a(n) for n = 0..78 (terms 0..30 from R. H. Hardin, terms 31..78 after A063443)

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).

A288956 Number of maximal independent vertex sets (and minimal vertex covers) in the n X n king graph.

Original entry on oeis.org

1, 4, 8, 79, 544, 8197, 201611, 6214593, 391918650, 32239887128, 4599025630995, 1018245217588836, 346578151637999287, 193445218205732588935, 165199496607694525364163, 226636538088997406396236072, 488063150616514603623041818756, 1655950305544572458601638523072809

Offset: 1

Eric W. Weisstein, Jun 20 2017

nonn

Eric Weisstein's World of Mathematics, King Graph
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Vertex Cover

Main diagonal of A332347.

Cf. A197048 (grid graph), A063443 (independent sets), A193580, A133791 (dominating sets).

a(9)-a(18) from Andrew Howroyd, Jun 26 2017

A337024 Number of ways to tile a 2n X 2n square with 1 X 1 white and n X n black squares.

Original entry on oeis.org

16, 35, 60, 91, 128, 171, 220, 275, 336, 403, 476, 555, 640, 731, 828, 931, 1040, 1155, 1276, 1403, 1536, 1675, 1820, 1971, 2128, 2291, 2460, 2635, 2816, 3003, 3196, 3395, 3600, 3811, 4028, 4251, 4480, 4715, 4956, 5203

Offset: 1

Yutong Li, Aug 11 2020

			For example, here are two of the 35 ways to tile a 4 X 4 square with 1 X 1 and 2 X 2 squares (where we have dropped the colors):
._______        _______
|_|_|   |      |_|_|   |
|   |___|      |_|_|___|
|___|   |      |   |   |
|_|_|___|      |_ _|___|

R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO], 2016, Section 4.1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A063443.

Table[3 n^2 + 10 n + 3, {n, 50}] (* Wesley Ivan Hurt, Nov 07 2020 *)

a(n) = 3*n^2 + 10*n + 3.
From Stefano Spezia, Aug 18 2020: (Start)
O.g.f.: x*(16 - 13*x + 3*x^2)/(1 - x)^3.
E.g.f.: exp(x)*(3 + 13*x + 3*x^2) - 3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

Edited by Greg Dresden, Aug 18 2020

A271081 Number of ways to tile an n X n X n cube with 1 X 1 X 1 and 2 X 2 X 2 tiles.

Original entry on oeis.org

1, 2, 9, 2089, 3144692, 2748613397101, 107008949868167431857

Offset: 1

Johan Nilsson, Mar 30 2016

			There are 9 ways to tile a cube of side length 3 with cubes of side length 1 and 2, so a(3) = 9.

Cf. A063443 (a 2D version of this tiling count).

cp's OEIS Frontend

A140314 Number of n X n binary matrices containing no more than two 1's in any 2 X 2 subblock.

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A182562 Number of ways to place k non-attacking semi-knights on an n x n chessboard, sum over all k>=0.

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A275869 Number of nonequivalent ways to place k>=0 nonattacking kings on an n X n board.

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A335560 Number of ways to tile an n X n square with 1 X 1 squares and (n-1) X 1 vertical or horizontal strips.

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A353934 Number of tilings of an n X n square using right trominoes, dominoes, and monominoes.

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A141483 Number of n X n binary matrices, symmetric under horizontal and vertical reflection, with no more than 1 one in any 2 X 2 subblock.

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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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A288956 Number of maximal independent vertex sets (and minimal vertex covers) in the n X n king graph.

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Author

Keywords

Links

Crossrefs

Extensions

A337024 Number of ways to tile a 2n X 2n square with 1 X 1 white and n X n black squares.

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Mathematica

Formula

Extensions

A271081 Number of ways to tile an n X n X n cube with 1 X 1 X 1 and 2 X 2 X 2 tiles.

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