A211154 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and even determinant.

1, 41, 457, 1345, 4481, 8521, 18985, 30017, 54721, 78121, 126281, 168961, 252097, 322505, 454441, 562561, 759425, 916777, 1197001, 1416641, 1800961, 2097481, 2608937, 2998465, 3662401, 4162601, 5006665, 5636737, 6690881, 7471561, 8768041, 9721601, 11294977, 12445225, 14332361, 15704641

Offset: 0

Author

Clark Kimberling, Apr 05 2012

Keywords

• nonn

Comments

For a guide to related sequences, see A210000.

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).

Crossrefs

Cf. A210000, A211155.

Programs

Maple

seq((2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n), n=1..20); # Mark van Hoeij, May 13 2013

Mathematica

a = -n; b = n; z1 = 20;
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, -2*n^2, 2*n^2}]
v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
Table[u[n], {n, 1, z1}] (* A211154 *)
Table[v[n], {n, 1, z1}] (* A211155 *)

PARI

a(n)=(2*n+1)^4 - 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n); \\ Joerg Arndt, May 14 2013

Formula

a(n) + A211155(n) = (2n+1)^4.

From Chai Wah Wu, Nov 27 2016: (Start)

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.

G.f.: x*(-x^8 - 36*x^6 - 416*x^5 - 734*x^4 - 1472*x^3 - 724*x^2 - 416*x - 41)/((x - 1)^5*(x + 1)^4). (End)

Extensions

More terms from Joerg Arndt, May 14 2013

a(0)=1 prepended by Andrew Howroyd, May 05 2020