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A210378 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even trace. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A210378(n)+A210379(n)=(n+1)^4.

See A210000 for a guide to related sequences.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..10000

FORMULA

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)

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.

Sequence in context: A118838 A153828 A247834 * A247534 A100648 A128091

Adjacent sequences:  A210375 A210376 A210377 * A210379 A210380 A210381

KEYWORD

nonn

AUTHOR

Clark Kimberling, Mar 20 2012

STATUS

approved

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Last modified June 25 00:41 EDT 2017. Contains 288708 sequences.