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A255876 a(n) = (4*n^2 + 4*n - 3 - 3*(-1)^n)/2.

%I A255876 #24 Dec 26 2024 18:48:06
%S A255876 4,9,24,37,60,81,112,141,180,217,264,309,364,417,480,541,612,681,760,
%T A255876 837,924,1009,1104,1197,1300,1401,1512,1621,1740,1857,1984,2109,2244,
%U A255876 2377,2520,2661,2812,2961,3120,3277,3444,3609,3784,3957,4140,4321,4512,4701
%N A255876 a(n) = (4*n^2 + 4*n - 3 - 3*(-1)^n)/2.
%C A255876 Take an n X n square grid and add unit squares along each side except for the corners --> do this repeatedly along each side with the same restriction until no squares can be added. a(n) gives the number of vertices in each figure (see example and cf. A255840).
%H A255876 Colin Barker, <a href="/A255876/b255876.txt">Table of n, a(n) for n = 1..1000</a>
%H A255876 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F A255876 G.f.: x*(3*x^3 - 6*x^2 - x - 4)/((x + 1)*(x - 1)^3).
%F A255876 a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
%F A255876 a(n) = A000290(n+1) + 4*A198442(n).
%e A255876                                                                  _
%e A255876                                                                _|_|_
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%e A255876                                                                 |_|
%e A255876    n=1        n=2          n=3             n=4                  n=5
%p A255876 A255876:=n->(4*n^2 + 4*n - 3 - 3*(-1)^n)/2: seq(A255876(n), n=1..50);
%t A255876 CoefficientList[Series[(3 x^3 - 6 x^2 - x - 4)/((x + 1) (x - 1)^3), {x, 0, 50}], x]
%t A255876 LinearRecurrence[{2,0,-2,1},{4,9,24,37},60] (* _Harvey P. Dale_, Dec 26 2024 *)
%o A255876 (Magma) [(4*n^2 + 4*n - 3 - 3*(-1)^n)/2 : n in [1..50]];
%o A255876 (PARI) vector(100,n,(4*n^2 + 4*n - 3 - 3*(-1)^n)/2) \\ _Derek Orr_, Mar 09 2015
%Y A255876 Cf. A000290 (squares), A085046, A198442, A255840.
%K A255876 nonn,easy
%O A255876 1,1
%A A255876 _Wesley Ivan Hurt_, Mar 08 2015