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

0,2

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) + 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)

Example

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

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, A210378.

See A210000 for a guide to related sequences.

Sequence in context: A341222 A213581 A276279 * A131123 A055910 A022573

Adjacent sequences: A210376 A210377 A210378 * A210380 A210381 A210382

Keyword

nonn

Author

Clark Kimberling, Mar 20 2012

Status

approved