%I #72 Sep 08 2022 08:46:21

%S 1,4,15,36,73,128,207,312,449,620,831,1084,1385,1736,2143,2608,3137,

%T 3732,4399,5140,5961,6864,7855,8936,10113,11388,12767,14252,15849,

%U 17560,19391,21344,23425,25636,27983,30468,33097,35872,38799,41880,45121,48524,52095

%N a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n) mod 2))/6.

%C a(n) is the trace of an n X n matrix A in which the entries are 1 through n^2, spiraling inward starting with 1 in the (1,1)-entry (proved).

%C The first three terms of a(n) coincide with those of A317614.

%H Stefano Spezia, <a href="/A304487/b304487.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).

%F a(n) = A045991(n) - Sum_{k=2..n-1} A085046(k) for n > 2 (proved).

%F G.f.: x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)).

%F a(n) + a(n + 1) = A228958(2*n + 1).

%F From _Colin Barker_, Aug 17 2018: (Start)

%F a(n) = (2*n - 3*n^2 + 4*n^3) / 6 for n even.

%F a(n) = (3 + 2*n - 3*n^2 + 4*n^3) / 6 for n odd.

%F a(n) = 3*a(n - 1) - 2*a(n - 2) - 2*a(n - 3) + 3*a(n - 4) - a(n - 5) for n > 5.

%F (End)

%F E.g.f.: (1/12)*exp(-x)*(-3 + exp(2*x)*(3 + 6*x + 18*x^2 + 8*x^3)). - _Stefano Spezia_, Feb 10 2019

%e For n = 1 the matrix A is

%e 1

%e with trace Tr(A) = a(1) = 1.

%e For n = 2 the matrix A is

%e 1, 2

%e 4, 3

%e with Tr(A) = a(2) = 4.

%e For n = 3 the matrix A is

%e 1, 2, 3

%e 8, 9, 4

%e 7, 6, 5

%e with Tr(A) = a(3) = 15.

%e For n = 4 the matrix A is

%e 1, 2, 3, 4

%e 12, 13, 14, 5

%e 11, 16, 15, 6

%e 10, 9, 8, 7

%e with Tr(A) = a(4) = 36.

%p seq((3+2*n-3*n^2+4*n^3-3*modp((-1+n),2))/6,n=1..43); # _Muniru A Asiru_, Sep 17 2018

%t Table[1/6 (3 + 2 n - 3 n^2 + 4 n^3 - 3 Mod[-1 + n, 2]), {n, 1, 43}] (* or *)

%t CoefficientList[ Series[x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)), {x, 0, 43}], x] (* or *)

%t LinearRecurrence[{3, -2, -2, 3, -1}, {1, 4, 15, 36, 73}, 43]

%o (MATLAB and FreeMat)

%o for(n=1:43); tm=(3 + 2*n - 3*n^2 + 4*n^3 - 3*mod(-1 + n, 2))/6; fprintf('%d\t%0.f\n', n, tm); end

%o (GAP)

%o a_n:=List([1..43], n->(3 + 2*n - 3*n^2 + 4*n^3 - 3*RemInt(-1 + n, 2))/6

%o (Maxima)

%o a(n):=(3 + 2*n - 3*n^2 + 4*n^3 - 3*mod(-1 + n, 2))/6$ makelist(a(n), n, 1, 43);

%o (PARI) Vec(x*(1 + x + 5*x^2 + x^3)/((-1 + x)^4*(1 + x)) + O(x^44))

%o (PARI) a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n)%2))/6

%o (Magma) I:=[1,4,15,36,73]; [n le 5 select I[n] else 3*Self(n-1)-2*Self(n-2)-2*Self(n-3)+3*Self(n-4)-Self(n-5): n in [1..43]]; // _Vincenzo Librandi_, Aug 26 2018

%o (GAP) List([1..43],n->(3+2*n-3*n^2+4*n^3-3*((-1+n) mod 2))/6); # _Muniru A Asiru_, Sep 17 2018

%Y Cf. A317614, A228958, A045991, A085046.

%Y Cf. A126224 (determinant of the matrix A), A317298 (first differences).

%K nonn,easy

%O 1,2

%A _Stefano Spezia_, Aug 17 2018