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A210379
Number of 2 X 2 matrices with all terms in {0,1,...,n} and odd trace.
4
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
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
See A210000 for a guide to related sequences.
Sequence in context: A341222 A213581 A276279 * A131123 A055910 A022573
KEYWORD
nonn
AUTHOR
Clark Kimberling, Mar 20 2012
STATUS
approved