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 1-5 of 5 results.

A162676 Number of different fixed (possibly) disconnected n-ominoes bounded (not necessarily tightly) by an n*n square.

Original entry on oeis.org

1, 4, 48, 956, 26490, 937342, 40291608, 2036155284, 118202408622, 7747410899954, 565695467415936, 45525704815717568, 4002930269944724664, 381750656962687053108, 39244733577786624617904, 4325973539461955182836900, 508971415418900757219557142
Offset: 1

Views

Author

David Bevan, Jul 27 2009

Keywords

Examples

			a(2)=4: the two rotations of the (connected) domino and the two rotations of the disconnected domino consisting of two squares connected at a vertex.

Crossrefs

Cf. A162673, A162674, A162675, A162677.

Programs

  • Mathematica
    Table[Binomial[n^2,n]-2*Binomial[(n-1)n,n]+Binomial[(n-1)^2,n],{n,20}] (* Harvey P. Dale, Oct 01 2013 *)
  • PARI
    a(n) = binomial(n^2,n) - 2*binomial((n-1)*n,n) + binomial((n-1)^2,n); \\ Michel Marcus, Aug 30 2013

Formula

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

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]

A163437 Number of different fixed (possibly) disconnected polyominoes (of any area) bounded tightly by an n X n square.

Original entry on oeis.org

1, 7, 322, 51472, 29671936, 64588152832, 545697103347712, 18161310923858378752, 2399054119350722118025216, 1262710910458264839283982467072, 2653270028014955753823799266500411392
Offset: 1

Views

Author

David Bevan, Jul 28 2009

Keywords

Examples

			a(2)=7: 2 rotations of the strictly disconnected domino consisting of two squares connected at a vertex, 4 rotations of the L tromino, and the square tetromino.

Links

Crossrefs

Cf. A162677 (bound not necessarily tight), A163433 (fixed disconnected trominoes), A163434 (fixed disconnected tetrominoes), A163435 (fixed disconnected pentominoes), A163436 (fixed disconnected n-ominoes).

Programs

  • Mathematica
    Table[2^(n^2) - 4*2^((n - 1)*n) + 4*2^((n - 1)^2) + 2*2^((n - 2)*n) -
  4*2^((n - 2)*(n - 1)) + 2^((n - 2)^2), {n, 1, 25}] (* G. C. Greubel, Dec 23 2016 *)

Formula

a(n) = 2^(n^2) - 4*2^((n-1)*n) + 4*2^((n-1)^2) + 2*2^((n-2)*n) - 4*2^((n-2)*(n-1)) + 2^((n-2)^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
Showing 1-5 of 5 results.

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