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

%I A163433 #45 Feb 16 2025 08:33:11
%S A163433 0,4,22,52,94,148,214,292,382,484,598,724,862,1012,1174,1348,1534,
%T A163433 1732,1942,2164,2398,2644,2902,3172,3454,3748,4054,4372,4702,5044,
%U A163433 5398,5764,6142,6532,6934,7348,7774,8212,8662,9124,9598,10084,10582,11092,11614
%N A163433 Number of different fixed (possibly) disconnected trominoes bounded tightly by an n X n square.
%C A163433 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
%C A163433 For n > 2, the surface area of a rectangular prism with sides n-2, n-1, and n. - _J. M. Bergot_, Sep 12 2011
%C A163433 Also the number of 4-cycles in the (n+2) X (n+2) knight graph. - _Eric W. Weisstein_, May 05 2017
%H A163433 G. C. Greubel, <a href="/A163433/b163433.txt">Table of n, a(n) for n = 1..1000</a>
%H A163433 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>.
%H A163433 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnightGraph.html">Knight Graph</a>.
%H A163433 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A163433 a(n) = 6*n^2 - 12*n + 4, n > 1.
%F A163433 From _Colin Barker_, Sep 06 2013: (Start)
%F A163433 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
%F A163433 G.f.: 2*x^2*(x^2-5*x-2) / (x-1)^3. (End)
%F A163433 a(n+1) = (n*i-1)^3 - (n*i+1)^3, where n > 0, i=sqrt(-1). - _Bruno Berselli_, Jan 23 2014
%F A163433 E.g.f.: 2*((3*x^2 - 3*x + 2)*exp(x) + x - 2). - _G. C. Greubel_, Dec 23 2016
%F A163433 From _Amiram Eldar_, Aug 20 2022: (Start)
%F A163433 Sum_{n>=2} 1/a(n) = 1/4 - cot(Pi/sqrt(3))*Pi/(4*sqrt(3)).
%F A163433 Sum_{n>=2} (-1)^n/a(n) = cosec(Pi/sqrt(3))*Pi/(4*sqrt(3)) - 1/4. (End)
%e A163433 a(2)=4: the four rotations of the (connected) L tromino.
%p A163433 A163433:=n->6*n^2 - 12*n + 4: 0,seq(A163433(n), n=2..100); # _Wesley Ivan Hurt_, May 05 2017
%t A163433 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 *)
%t A163433 Join[{0}, Table[6*n^2 - 12*n + 4, {n, 2, 50}]] (* _G. C. Greubel_, Dec 23 2016 *)
%t A163433 Join[{0}, LinearRecurrence[{3, -3, 1}, {4, 22, 52}, 50]] (* _G. C. Greubel_, Dec 23 2016 *)
%t A163433 Length /@ Table[FindCycle[KnightTourGraph[n + 2, n + 2], {4}, All], {n, 20}] (* _Eric W. Weisstein_, May 05 2017 *)
%o A163433 (PARI) concat([0], Vec(2*x^2*(x^2-5*x-2) / (x-1)^3 + O(x^50))) \\ _G. C. Greubel_, Dec 23 2016
%Y A163433 Cf. A162673, A163434, A163437.
%Y A163433 Cf. A289181 (6-cycles in the n X n knight graph).
%K A163433 nonn,easy
%O A163433 1,2
%A A163433 _David Bevan_, Jul 28 2009