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

0,2

Comments

A211154(n) + A211155(n) = (2n+1)^4.

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).

Formula

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)

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 *)

Prog

(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

Crossrefs

Cf. A210000, A211155.

Sequence in context: A173768 A061643 A209842 * A103735 A177491 A269087

Adjacent sequences: A211151 A211152 A211153 * A211155 A211156 A211157

Keyword

nonn

Author

Clark Kimberling, Apr 05 2012

Extensions

More terms from Joerg Arndt, May 14 2013

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

Status

approved