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A280058
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Number of 2 X 2 matrices with entries in {0,1,...,n} with determinant = permanent with no entries repeated.
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1
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0, 0, 0, 12, 48, 120, 240, 420, 672, 1008, 1440, 1980, 2640, 3432, 4368, 5460, 6720, 8160, 9792, 11628, 13680, 15960, 18480, 21252, 24288, 27600, 31200, 35100, 39312, 43848, 48720, 53940, 59520, 65472, 71808, 78540, 85680, 93240, 101232, 109668, 118560
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OFFSET
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0,4
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COMMENTS
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Consider all Pythagorean triples (X,Y,Z=Y+2) ordered by increasing Z; A005843, A005563, A002522 and A007531 give the X, Y, Z and area A values of related triangles; for n >= 2 altitude h(n) = a(n+1)/A002522(n) or h(n)/2 is irreducible fraction in Q\Z. - Ralf Steiner, Mar 29 2020
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LINKS
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FORMULA
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a(n) = 2*((n+1)^3 - 6*(n+1)^2 + 11*(n+1) - 6), for n>0.
a(n) == 12 (mod 12).
G.f.: (12*x^3)/(1 - x)^4.
E.g.f.: 2*x^3*exp(x).
a(n) = 2*n*(n-1)*(n-2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)
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MATHEMATICA
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Table[2*n*(n-1)*(n-2), {n, 0, 50}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {0, 0, 0, 12}, 50] (* G. C. Greubel, Dec 25 2016 *)
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PROG
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(Python)
def t(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-b*c)==(a*d+b*c):
s+=1
return s
for i in range(0, 201):
print str(i)+" "+str(t(i))
(PARI) for(n=0, 50, print1(2*n*(n-1)*(n-2), ", ")) \\ G. C. Greubel, Dec 25 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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