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).

A163433 Number of different fixed (possibly) disconnected trominoes bounded tightly by an n X n square.

Original entry on oeis.org

0, 4, 22, 52, 94, 148, 214, 292, 382, 484, 598, 724, 862, 1012, 1174, 1348, 1534, 1732, 1942, 2164, 2398, 2644, 2902, 3172, 3454, 3748, 4054, 4372, 4702, 5044, 5398, 5764, 6142, 6532, 6934, 7348, 7774, 8212, 8662, 9124, 9598, 10084, 10582, 11092, 11614
Offset: 1

Views

Author

David Bevan, Jul 28 2009

Keywords

Comments

Except for the first term of 0, a(n) is the set of all integers k such that 6k+12 is a perfect square. - Gary Detlefs, Mar 01 2010
For n > 2, the surface area of a rectangular prism with sides n-2, n-1, and n. - J. M. Bergot, Sep 12 2011
Also the number of 4-cycles in the (n+2) X (n+2) knight graph. - Eric W. Weisstein, May 05 2017

Examples

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

Links

Crossrefs

Cf. A162673, A163434, A163437.
Cf. A289181 (6-cycles in the n X n knight graph).

Programs

  • Maple
    A163433:=n->6*n^2 - 12*n + 4: 0,seq(A163433(n), n=2..100); # Wesley Ivan Hurt, May 05 2017
  • Mathematica
    CoefficientList[Series[(2*z*(z^3 - 5*z^2 - 2*z))/(z - 1)^3, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)
Join[{0}, Table[6*n^2 - 12*n + 4, {n, 2, 50}]] (* G. C. Greubel, Dec 23 2016 *)
Join[{0}, LinearRecurrence[{3, -3, 1}, {4, 22, 52}, 50]] (* G. C. Greubel, Dec 23 2016 *)
Length /@ Table[FindCycle[KnightTourGraph[n + 2, n + 2], {4}, All], {n, 20}] (* Eric W. Weisstein, May 05 2017 *)
  • PARI
    concat([0], Vec(2*x^2*(x^2-5*x-2) / (x-1)^3 + O(x^50))) \\ G. C. Greubel, Dec 23 2016

Formula

a(n) = 6*n^2 - 12*n + 4, n > 1.
From Colin Barker, Sep 06 2013: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
G.f.: 2*x^2*(x^2-5*x-2) / (x-1)^3. (End)
a(n+1) = (n*i-1)^3 - (n*i+1)^3, where n > 0, i=sqrt(-1). - Bruno Berselli, Jan 23 2014
E.g.f.: 2*((3*x^2 - 3*x + 2)*exp(x) + x - 2). - G. C. Greubel, Dec 23 2016
From Amiram Eldar, Aug 20 2022: (Start)
Sum_{n>=2} 1/a(n) = 1/4 - cot(Pi/sqrt(3))*Pi/(4*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = cosec(Pi/sqrt(3))*Pi/(4*sqrt(3)) - 1/4. (End)

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).

A077958 Expansion of 1/(1-2*x^3).

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, 32, 0, 0, 64, 0, 0, 128, 0, 0, 256, 0, 0, 512, 0, 0, 1024, 0, 0, 2048, 0, 0, 4096, 0, 0, 8192, 0, 0, 16384, 0, 0, 32768, 0, 0, 65536, 0, 0, 131072, 0, 0, 262144, 0, 0, 524288, 0, 0, 1048576, 0, 0, 2097152, 0, 0, 4194304, 0, 0, 8388608, 0
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2002

Keywords

Comments

a(n) is the number of L-tromino tilings of the n X 2 rectangle (see Exercise 2 in Grinberg). - Stefano Spezia, Nov 26 2019

Links

Crossrefs

Cf. A006801, A162673, A163433, A329185.

Programs

Formula

From Stefano Spezia, Nov 26 2019: (Start)
a(n) = 2^(n/3) if 3 divides n, otherwise a(n) = 0 (see Exercise 2 in Grinberg).
E.g.f.: (1/3)*(exp(-(-2)^(1/3)*x) + exp(2^(1/3)*x) + exp((-1)^(2/3)*2^(1/3)*x)). (End)
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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