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A028875
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a(n) = n^2 - 5.
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13
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-5, -4, -1, 4, 11, 20, 31, 44, 59, 76, 95, 116, 139, 164, 191, 220, 251, 284, 319, 356, 395, 436, 479, 524, 571, 620, 671, 724, 779, 836, 895, 956, 1019, 1084, 1151, 1220, 1291, 1364, 1439, 1516, 1595, 1676, 1759, 1844, 1931, 2020, 2111, 2204, 2299, 2396
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OFFSET
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0,1
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COMMENTS
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a(n) gives the values for a*c of indefinite binary quadratic forms [a, b, c] of discriminant D = 20 for b = 2*n. In general D = b^2 - 4*a*c > 0 and the form [a, b, c] is a*x^2 + b*x*y + c*y^2. - Wolfdieter Lang, Aug 15 2013
For n>2, a(n) represents the area of the triangle created by the three points defined with coordinates: (n-3,n-2), ((n-1)*n/2,n*(n+1)/2), and ((n+1)^2, (n+2)^2). - J. M. Bergot, May 22 2014
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LINKS
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FORMULA
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G.f.: x^3*(4 - x - x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
Sum_{n>=0} 1/a(n) = -(1 + sqrt(5)*Pi*cot(sqrt(5)*Pi))/10.
Sum_{n>=0} (-1)^n/a(n) = -(1 + sqrt(5)*Pi*cosec(sqrt(5)*Pi))/10. (End)
Product_{n>=0} (1 - 1/a(n)) = sqrt(6/5)*sin(sqrt(6)*Pi)/sin(sqrt(5)*Pi).
Product_{n>=3} (1 + 1/a(n)) = sqrt(5)*Pi/(6*sin(sqrt(5)*Pi)). (End)
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MAPLE
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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