A210379 Number of 2 X 2 matrices with all terms in {0,1,...,n} and odd trace.

0, 8, 36, 128, 300, 648, 1176, 2048, 3240, 5000, 7260, 10368, 14196, 19208, 25200, 32768, 41616, 52488, 64980, 80000, 97020, 117128, 139656, 165888, 195000, 228488, 265356, 307328, 353220, 405000, 461280, 524288, 592416, 668168

Offset: 0

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn

Examples

			Writing the matrices as 4-letter words, the 8 for n=1 are as follows:
1000, 1100, 1010, 1110, 0001, 0011, 0101, 0111
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (2, 2, -6, 0, 6, -2, -2, 1).

Crossrefs

Cf. A210000, A210378.

See A210000 for a guide to related sequences.

Programs

Mathematica

a = 0; b = n; z1 = 35;
t[n_] := t[n] = Flatten[Table[w + z, {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, 0, 2*n}]
v[n_] := Sum[c[n, 2 k - 1], {k, 1, 2*n - 1}]
Table[u[n], {n, 0, z1}] (* A210378 *)
Table[v[n], {n, 0, z1}] (* A210379 *)

Formula

a(n) + A210378(n) = (n+1)^4.

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

a(n) = (n + 1)^2*((n + 1)^2 - (2*n + 1 -(-1)^n)^2/16 - (2*n + 3 + (-1)^n)^2/16).

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

G.f.: -4*x*(2*x^4 + 5*x^3 + 10*x^2 + 5*x + 2)/((x - 1)^5*(x + 1)^3). (End)

From Amiram Eldar, Mar 15 2024: (Start)

a(n) = (n+1)^2*floor((n+1)^2/2).

Sum_{n>=1} 1/a(n) = Pi^4/720 + (10-Pi^2)/4. (End)