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 1-4 of 4 results.

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)

A163434 Number of different fixed (possibly) disconnected tetrominoes bounded tightly by an n X n square.

Original entry on oeis.org

0, 1, 70, 425, 1426, 3577, 7526, 14065, 24130, 38801, 59302, 87001, 123410, 170185, 229126, 302177, 391426, 499105, 627590, 779401, 957202, 1163801, 1402150, 1675345, 1986626, 2339377, 2737126, 3183545, 3682450, 4237801, 4853702
Offset: 1

Views

Author

David Bevan, Jul 28 2009

Keywords

Examples

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

Links

Crossrefs

Cf. A162674, A163433, A163435, A163437.

Programs

  • Mathematica
    Join[{0}, Table[(2 n^2 - 4 n + 1)*(3 n^2 - 6 n + 1), {n, 2, 50}]] (* or *) Join[{0}, LinearRecurrence[{5,-10,10,-5,1}, {1, 70, 425, 1426, 3577}, 50]] (* G. C. Greubel, Dec 23 2016 *)
  • PARI
    concat([0], Vec(x^2*(1+65*x+85*x^2-9*x^3+2*x^4)/(1-x)^5 + O(x^50))) \\ G. C. Greubel, Dec 23 2016

Formula

a(n) = (2n^2 -4n +1)*(3n^2 -6n +1), n>1.
G.f.: x^2*(1+65*x+85*x^2-9*x^3+2*x^4)/(1-x)^5. - Colin Barker, Apr 25 2012
E.g.f.: (6*x^4 + 12*x^3 - x^2 + x + 1)*exp(x) - 2 x - 1. - G. C. Greubel, Dec 23 2016

A163435 Number of different fixed (possibly) disconnected pentominoes bounded tightly by an n X n square.

Original entry on oeis.org

0, 0, 102, 1792, 11550, 46848, 144550, 371712, 838782, 1715200, 3247398, 5779200, 9774622, 15843072, 24766950, 37531648, 55357950, 79736832, 112466662, 155692800, 211949598, 284204800, 375906342, 491031552, 634138750, 810421248
Offset: 1

Views

Author

David Bevan, Jul 28 2009

Keywords

Examples

			a(3) = 102: there are 102 rotations of the 19 free (possibly) disconnected pentominoes bounded tightly by a 3 X 3 square; these include the F, T, V, W, X and Z (connected) pentominoes and 13 strictly disconnected free pentominoes.

Links

Crossrefs

Cf. A162675, A163434, A163437.

Programs

  • Mathematica
    Join[{0}, Table[(2/3)*n^2*(n - 2)^2*(5*n^2 - 10*n + 2), {n, 2, 50}]] (* or *) Join[{0}, LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0, 102, 1792, 11550, 46848, 144550, 371712}, 50]] (* G. C. Greubel, Dec 23 2016 *)
  • PARI
    concat([0,0], Vec(2*x^3*(51+539*x+574*x^2+30*x^3+7*x^4-x^5)/ (1-x)^7 + O(x^50))) \\ G. C. Greubel, Dec 23 2016

Formula

a(n) = 2/3*n^2*(n-2)^2*(5*n^2-10*n+2), n>1.
G.f.: 2*x^3*(51+539*x+574*x^2+30*x^3+7*x^4-x^5)/(1-x)^7. - Colin Barker, Apr 25 2012
E.g.f.: (2/3)*x*(5*x^5 + 45*x^4 + 87*x^3 + 24*x^2 + 3*x - 3)*exp(x) + 2*x. - G. C. Greubel, Dec 23 2016

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.
Showing 1-4 of 4 results.

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