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

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

Author

David Bevan, Jul 28 2009

Keywords

• nonn
• easy

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

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Graph Cycle.
• Eric Weisstein's World of Mathematics, Knight Graph.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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)