A211155 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and odd determinant.

0, 40, 168, 1056, 2080, 6120, 9576, 20608, 28800, 52200, 68200, 110880, 138528, 208936, 252840, 360960, 426496, 583848, 677160, 896800, 1024800, 1321320, 1491688, 1881216, 2102400, 2602600, 2883816, 3513888, 3865120, 4645800, 5077800, 6031360, 6555648, 7705896, 8334760, 9707040

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*(-40*x^6 - 128*x^5 - 728*x^4 - 512*x^3 - 728*x^2 - 128*x - 40)/((x - 1)^5*(x + 1)^4). (End)

Maple

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

Crossrefs

Cf. A210000.

Sequence in context: A250585 A260169 A205249 * A250994 A250995 A262487

Adjacent sequences: A211152 A211153 A211154 * A211156 A211157 A211158

Keyword

nonn

Author

Clark Kimberling, Apr 05 2012

Extensions

More terms from Joerg Arndt, May 14 2013

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

Status

approved