A280056 Number of 2 X 2 matrices with entries in {0,1,...,n} and even trace with no entries repeated.
0, 0, 0, 8, 48, 144, 360, 720, 1344, 2240, 3600, 5400, 7920, 11088, 15288, 20384, 26880, 34560, 44064, 55080, 68400, 83600, 101640, 121968, 145728, 172224, 202800, 236600, 275184, 317520, 365400, 417600, 476160, 539648, 610368, 686664, 771120, 861840, 961704, 1068560, 1185600
Offset: 0
Links
- Indranil Ghosh, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).
Crossrefs
Cf. A210378 (where the elements can be repeated).
Programs
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Mathematica
Table[(1/4)*(n - 2)*(n - 1)*(2*n^2 - 1 + (-1)^n), {n, 0, 50}] (* G. C. Greubel, Dec 26 2016 *) -
PARI
concat(vector(3), Vec(8*x^3*(1 + 3*x)*(1 + x + x^2) / ((1 - x )^5*(1 + x)^3) + O(x^30))) \\ Colin Barker, Dec 25 2016
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Python
def a(n): s=0 for a in range(0,n+1): for b in range(0,n+1): if a!=b: for c in range(0,n+1): if a!=c and b!=c: for d in range(0,n+1): if d!=a and d!=b and d!=c: if (a+d)%2==0: s+=1 return s for i in range(0,41): print(i, a(i)) -
Python
def A280056(n): return (n**2 - (n % 2))*(n-1)*(n-2)//2 # Chai Wah Wu, Dec 25 2016
Formula
a(n) = ((-1)^n + 2*n^2 - 1)*(n-1)*(n-2)/4.
From Colin Barker, Dec 25 2016: (Start)
a(n) = (n^4 - 3*n^3 + 2*n^2)/2 for n even.
a(n) = (n^4 - 3*n^3 + n^2 + 3*n - 2)/2 for n odd.
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.: 8*x^3*(1 + 3*x)*(1 + x + x^2) / ((1 - x )^5*(1 + x)^3). (End)
These formulas are true. a(n) = ((-1)^n + 2*n^2 - 1)*(n-1)*(n-2)/4 = (n^2 - p(n))*C(n-1,2), where p(n) is the parity of n, i.e., p(n) = 0 if n is even and p(n) = 1 if n is odd. - Chai Wah Wu, Dec 25 2016
E.g.f.: (1/4)*((2 + 2*x + x^2)*exp(-x) + (-2 + 2*x - x^2 + 6*x^3 + 2*x^4)*exp(x)). - David Radcliffe, Aug 16 2025
Extensions
Formulas corrected by David Radcliffe, Aug 16 2025
Comments