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A210378 Number of 2 X 2 matrices with all terms in {0,1,...,n} and even trace. 4

%I

%S 1,8,45,128,325,648,1225,2048,3321,5000,7381,10368,14365,19208,25425,

%T 32768,41905,52488,65341,80000,97461,117128,140185,165888,195625,

%U 228488,266085,307328,354061,405000,462241,524288,593505,668168

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

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

%C See A210000 for a guide to related sequences.

%H Chai Wah Wu, <a href="/A210378/b210378.txt">Table of n, a(n) for n = 0..10000</a>

%F From _Chai Wah Wu_, Nov 27 2016: (Start)

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

%F 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.

%F 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)

%e Writing the matrices as 4-letter words, the 8 for n=1 are as follows:

%e 0000, 0100, 0010, 0110, 1001, 1101, 1011, 1111

%t a = 0; b = n; z1 = 35;

%t t[n_] := t[n] = Flatten[Table[w + z, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]

%t c[n_, k_] := c[n, k] = Count[t[n], k]

%t u[n_] := Sum[c[n, 2 k], {k, 0, 2*n}]

%t v[n_] := Sum[c[n, 2 k - 1], {k, 1, 2*n - 1}]

%t Table[u[n], {n, 0, z1}] (* A210378 *)

%t Table[v[n], {n, 0, z1}] (* A210379 *)

%Y Cf. A210000, A210379.

%K nonn

%O 0,2

%A _Clark Kimberling_, Mar 20 2012

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Last modified November 13 00:25 EST 2019. Contains 329083 sequences. (Running on oeis4.)