A304487 a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n) mod 2))/6.

1, 4, 15, 36, 73, 128, 207, 312, 449, 620, 831, 1084, 1385, 1736, 2143, 2608, 3137, 3732, 4399, 5140, 5961, 6864, 7855, 8936, 10113, 11388, 12767, 14252, 15849, 17560, 19391, 21344, 23425, 25636, 27983, 30468, 33097, 35872, 38799, 41880, 45121, 48524, 52095

Offset

1,2

Comments

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).

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

Links

Stefano Spezia, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Formula

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

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

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

From Colin Barker, Aug 17 2018: (Start)

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

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

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

(End)

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

Example

For n = 1 the matrix A is
   1
with trace Tr(A) = a(1) = 1.
For n = 2 the matrix A is
   1, 2
   4, 3
with Tr(A) = a(2) = 4.
For n = 3 the matrix A is
   1, 2, 3
   8, 9, 4
   7, 6, 5
with Tr(A) = a(3) = 15.
For n = 4 the matrix A is
   1,  2,  3, 4
  12, 13, 14, 5
  11, 16, 15, 6
  10,  9,  8, 7
with Tr(A) = a(4) = 36.

Maple

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

Mathematica

Table[1/6 (3 + 2 n - 3 n^2 + 4 n^3 - 3 Mod[-1 + n, 2]), {n, 1, 43}] (* or *)
CoefficientList[ Series[x*(1 + x + 5 x^2 + x^3)/((-1 + x)^4 (1 + x)), {x, 0, 43}], x] (* or *)
LinearRecurrence[{3, -2, -2, 3, -1}, {1, 4, 15, 36, 73}, 43]

Prog

(MATLAB and FreeMat)
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
(GAP)
a_n:=List([1..43], n->(3 + 2*n - 3*n^2 + 4*n^3 - 3*RemInt(-1 + n, 2))/6
(Maxima)
a(n):=(3 + 2*n - 3*n^2 + 4*n^3 - 3*mod(-1 + n, 2))/6$ makelist(a(n), n, 1, 43);
(PARI) Vec(x*(1 + x + 5*x^2 + x^3)/((-1 + x)^4*(1 + x)) + O(x^44))
(PARI) a(n) = (3 + 2*n - 3*n^2 + 4*n^3 - 3*((-1 + n)%2))/6
(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
(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

Crossrefs

Cf. A317614, A228958, A045991, A085046.

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

Sequence in context: A330204 A190093 A077414 * A350689 A015653 A106199

Adjacent sequences: A304484 A304485 A304486 * A304488 A304489 A304490

Keyword

nonn,easy

Author

Stefano Spezia, Aug 17 2018

Status

approved