cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A162673 Number of different fixed (possibly) disconnected trominoes bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

0, 4, 48, 204, 580, 1320, 2604, 4648, 7704, 12060, 18040, 26004, 36348, 49504, 65940, 86160, 110704, 140148, 175104, 216220, 264180, 319704, 383548, 456504, 539400, 633100, 738504, 856548, 988204, 1134480, 1296420, 1475104, 1671648, 1887204
Offset: 1

Views

Author

David Bevan, Jul 27 2009

Keywords

Comments

Fixed quasi-trominoes.

Examples

			a(2)=4: the four rotations of the (connected) L tromino

Links

Crossrefs

Cf. A162674, A162676, A162677, A094170 (free quasi-trominoes).

Programs

  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1},{0,4,48,204,580},40] (* Harvey P. Dale, Aug 09 2017 *)

Formula

a(n)=n*(n-1)*(3*n^2-3*n-2)/2.
G.f.: 4*x^2*(1+7*x+x^2)/(1-x)^5. [Colin Barker, Apr 25 2012]

A162674 Number of different fixed (possibly) disconnected tetrominoes bounded (not necessarily tightly) by an n X n square.

Original entry on oeis.org

0, 1, 97, 956, 4780, 16745, 46921, 112672, 241536, 474585, 870265, 1508716, 2496572, 3972241, 6111665, 9134560, 13311136, 18969297, 26502321, 36377020, 49142380, 65438681, 86007097, 111699776, 143490400, 182485225, 229934601
Offset: 1

Views

Author

David Bevan, Jul 27 2009

Keywords

Comments

Fixed quasi-tetrominoes.

Examples

			a(2)=1: the (connected) square tetromino.

Links

Crossrefs

Cf. A162673, A162675, A162676, A162677, A094171 (free quasi-tetrominoes).

Formula

a(n) = n*(n-1)*(8*n^4-16*n^3-9*n^2+17*n+8)/12.
G.f.: x^2*(1+90*x+298*x^2+90*x^3+x^4)/(1-x)^7. [Colin Barker, Apr 25 2012]

A162677 Number of different fixed (possibly) disconnected polyominoes (of any area) bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

1, 10, 400, 57856, 31522816, 66605547520, 554222579875840, 18303191835587117056, 2408425353007592768536576, 1265177138001297870205254369280, 2655861110791164560222750369099284480
Offset: 1

Views

Author

David Bevan, Jul 27 2009

Keywords

Examples

			a(2)=10: the monomino, 4 dominoes (2 strictly disconnected), 4 rotations of the L tromino, and the square tetromino.

Crossrefs

Cf. A162673, A162674, A162675, A162676.

Programs

  • PARI
    a(n) = 2^(n^2) - 2*2^((n-1)*n) + 2^((n-1)^2); \\ Michel Marcus, Aug 30 2013

Formula

a(n) = 2^(n^2)-2*2^((n-1)*n)+2^((n-1)^2).

A162675 Number of different fixed (possibly) disconnected pentominoes bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

0, 0, 114, 2910, 26490, 145110, 582540, 1891764, 5263020, 13010580, 29297070, 61162530, 119933814, 223098330, 396734520, 678599880, 1121985720, 1800456264, 2813598090, 4293914310, 6415006290, 9401194110, 13538735364, 19188810300
Offset: 1

Views

Author

David Bevan, Jul 27 2009

Keywords

Comments

Fixed quasi-pentominoes.

Examples

			a(3)=114: there are 114 rotations of the 21 free (possibly) disconnected pentominoes bounded (not necessarily tightly) by an 3*3 square; these include the F, P, T, U, V, W, X and Z (connected) pentominoes and 13 strictly disconnected pentominoes.

Links

Crossrefs

Cf. A162674, A162676, A162677, A094172 (free quasi-pentominoes).

Formula

a(n) = n*(n-1)*(n-2)*(n+1)*(5*n^4-10*n^3-7*n^2+12*n+6)/24.
G.f.: x^3*(114+1884*x+4404*x^2+1884*x^3+114*x^4)/(1-x)^9. [Colin Barker, Apr 25 2012]

Extensions

Example moved to correct section, and ref to free quasi-pentominoes added by David Bevan, Mar 05 2011

A163436 Number of different fixed (possibly) disconnected n-ominoes bounded tightly by an n*n square.

Original entry on oeis.org

1, 2, 22, 425, 11550, 403252, 17164532, 860938920, 49684113582, 3240906864140, 235707022877304, 18906047682170948, 1657638292334575486, 157698852357527675040, 16177213677228994535040, 1779883643542856425993296, 209064002262265290212455374
Offset: 1

Views

Author

David Bevan, Jul 28 2009

Keywords

Examples

			a(2)=2: the two rotations of the strictly disconnected domino consisting of two squares connected at a vertex

Links

Crossrefs

Cf. A162676, A163437

Programs

  • Magma
    [1] cat [Binomial(n^2, n)-4*Binomial((n-1)*n, n)+ 4*Binomial((n-1)^2, n)+2*Binomial((n-2)*n, n)-4*Binomial((n- 2)*(n-1), n)+Binomial((n-2)^2, n): n in [2..20]]; // Vincenzo Librandi, Dec 23 2016
  • Mathematica
    Join[{1}, Table[Binomial[n^2, n] - 4*Binomial[(n - 1)*n, n] + 4*Binomial[(n - 1)^2, n] + 2*Binomial[(n - 2)*n, n] - 4*Binomial[(n - 2)*(n - 1), n] + Binomial[(n - 2)^2, n], {n, 2, 50}]] (* G. C. Greubel, Dec 23 2016 *)

Formula

a(n)=binomial(n^2,n)-4*binomial((n-1)*n,n)+4*binomial((n-1)^2,n)+2*binomial((n-2)*n,n)-4*binomial((n-2)*(n-1),n)+binomial((n-2)^2,n), n>1.

A272435 Number of n-ominoes in n X n grid (i.e., rookwise connected sets of n cells in a square array with n rows and n columns).

Original entry on oeis.org

1, 4, 22, 113, 571, 2816, 13616, 64678, 302574, 1397318, 6382660, 28882214, 129640058, 577812724, 2559491834, 11276000877, 49437494408, 215815377168
Offset: 1

Views

Author

Don Knuth, Apr 29 2016

Keywords

Comments

Higher corrected values are supported by exhibiting a(n) distinct n-ominoes in the n-square for n=10 and n=11 (see LINKS below). - James Stein, Dec 11 2017
a(n) is the number of connected induced subgraphs with n vertices in the n X n grid graph. - Andrew Howroyd, Apr 27 2020

Examples

			The 22 arrangements for n=3 include three horizontal rows, three vertical rows, and four ways to place each rotation of the L-tromino.

References

  • This sequence will some day be mentioned in an exercise in section 7.2.2 of The Art of Computer Programming.

Links

Crossrefs

Cf. A001168, A059525, A162676.

Extensions

a(10)-a(12) corrected, and a(13)-a(14) added by James Stein, Dec 11 2017
a(15)-a(16) from Andrew Howroyd, Apr 27 2020
a(17)-a(18) from Giovanni Resta, May 01 2020

A162678 Number of fixed strictly disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

0, 2, 42, 937, 26427, 937126, 40290848, 2036152559, 118202398712, 7747410863508, 565695467280668, 45525704815211707, 4002930269942820774, 381750656962679848234, 39244733577786597223238
Offset: 1

Views

Author

David Bevan, Jul 28 2009

Keywords

Comments

a(n) = A162676(n) - A001168(n)

Examples

			a(2)=2: the two rotations of the disconnected domino consisting of two squares connected at a vertex

Crossrefs

Cf. A162676, A001168, A006770, A030222
Showing 1-7 of 7 results.

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