OFFSET
0,2
COMMENTS
a(n) is also the number of 2 X 2 matrices with all terms in {0,1,...n} and even permanent.
The determinant will be even if either all entries are odd or if both the leading and trailing diagonals have no more than one odd entry each. - Andrew Howroyd, Apr 28 2020
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, 4, -4, -6, 6, 4, -4, -1, 1).
FORMULA
a(n) + A210370(n) = n^4.
From Colin Barker, Nov 28 2014: (Start)
a(n) = (13 + 3*(-1)^n + 4*(13+3*(-1)^n)*n + 2*(37+7*(-1)^n)*n^2 + 4*(11+(-1)^n)*n^3 + 10*n^4)/16.
G.f.: -(x^7+9*x^6+27*x^5+83*x^4+59*x^3+51*x^2+9*x+1) / ((x-1)^5*(x+1)^4).
(End)
a(n) = ((n+1)^2 - ceiling(n/2)^2)^2 + ceiling(n/2)^4. - Andrew Howroyd, Apr 28 2020
MATHEMATICA
a = 0; b = n; z1 = 28;
t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
c[n_, k_] := c[n, k] = Count[t[n], k]
u[n_] := Sum[c[n, 2 k], {k, -n^2, n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -n^2, n^2}]
Table[u[n], {n, 0, z1}] (* A210369 *)
Table[v[n], {n, 0, z1}] (* A210370 *)
PROG
(PARI) a(n) = {((n+1)^2 - ceil(n/2)^2)^2 + ceil(n/2)^4} \\ Andrew Howroyd, Apr 28 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 20 2012
EXTENSIONS
Terms a(29) and beyond from Andrew Howroyd, Apr 28 2020
STATUS
approved