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A325657 a(n) = (1/2)*(-1 + (-1)^n)*(n-1) + n^2.

%I A325657 #7 Sep 08 2022 08:46:24
%S A325657 0,1,4,7,16,21,36,43,64,73,100,111,144,157,196,211,256,273,324,343,
%T A325657 400,421,484,507,576,601,676,703,784,813,900,931,1024,1057,1156,1191,
%U A325657 1296,1333,1444,1483,1600,1641,1764,1807,1936,1981,2116,2163,2304,2353,2500,2551
%N A325657 a(n) = (1/2)*(-1 + (-1)^n)*(n-1) + n^2.
%C A325657 For n > 0, a(n) is the n-th element of the diagonal of the triangle A325655. Equivalently, a(n) is the element M_{n,1} of the matrix M(n) whose permanent is A322277(n).
%H A325657 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A325657 O.g.f.: (-1 - 3*x - x^2 - 3*x^3)/((-1 + x)^3*(1+x)^2).
%F A325657 E.g.f.: (1/2)*exp(-x)*(-1 - x + exp(2*x)*(1 + x + 2*x^2)).
%F A325657 a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4
%F A325657 a(n) = n^2 if n is even.
%F A325657 a(n) = n^2 - n + 1 if n is odd.
%p A325657 a:=n->(1/24)*n*(3 - 3*(- 1)^n + 4*n + 6*n^2 + 8*n^3): seq(a(n), n=0..55);
%t A325657 Table[(1/2)*(- 1+(-1)^n)*(n-1)+n^2,{n,0,55}]
%o A325657 (GAP) Flat(List([0..55], n->(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2));
%o A325657 (Magma) [(1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2: n in [0..55]];
%o A325657 (PARI) a(n) = (1/2)*(- 1 + (- 1)^n)*(n - 1) + n^2;
%Y A325657 Cf. A000290, A054569, A317614, A322277, A325655.
%K A325657 nonn,easy
%O A325657 0,3
%A A325657 _Stefano Spezia_, May 13 2019