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

0,2

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).

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

Example

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

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 *)

Crossrefs

Cf. A210000, A210379.

See A210000 for a guide to related sequences.

Sequence in context: A118838 A153828 A247834 * A247534 A100648 A128091

Adjacent sequences: A210375 A210376 A210377 * A210379 A210380 A210381

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2012

Status

approved