A210378 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even trace.

1, 8, 45, 128, 325, 648, 1225, 2048, 3321, 5000, 7381, 10368, 14365, 19208, 25425, 32768, 41905, 52488, 65341, 80000, 97461, 117128, 140185, 165888, 195625, 228488, 266085, 307328, 354061, 405000, 462241, 524288, 593505, 668168, 750925, 839808, 937765, 1042568

Offset: 0

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn
• easy

Examples

			Writing the matrices as 4-letter words, the 8 for n=1 are as follows:
0000, 0100, 0010, 0110, 1001, 1101, 1011, 1111
		

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, A210379.

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) + A210379(n) = (n+1)^4.

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

a(n) = (n + 1)^2*((2*n + 1 -(-1)^n)^2 + (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.: (-x^6 - 6*x^5 - 27*x^4 - 28*x^3 - 27*x^2 - 6*x - 1)/((x - 1)^5*(x + 1)^3). (End)

From Amiram Eldar, Mar 15 2024: (Start)

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

Sum_{n>=0} 1/a(n) = Pi^4/720 + (Pi-2*tanh(Pi/2))*Pi/4. (End)

E.g.f.: ((2 + 15*x + 26*x^2 + 10*x^3 + x^4)*cosh(x) + (1 + 18*x + 25*x^2 + 10*x^3 + x^4)*sinh(x))/2. - Stefano Spezia, Jul 15 2024