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A028347
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a(n) = n^2 - 4.
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71
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0, 5, 12, 21, 32, 45, 60, 77, 96, 117, 140, 165, 192, 221, 252, 285, 320, 357, 396, 437, 480, 525, 572, 621, 672, 725, 780, 837, 896, 957, 1020, 1085, 1152, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 2205, 2300, 2397, 2496, 2597
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OFFSET
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2,2
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COMMENTS
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Nonnegative X values of solutions to the equation X^3 + 4*X^2 = Y^2. The respective Y values are n*(n^2 - 4). - Mohamed Bouhamida, Nov 06 2007
Discriminants of binary forms x^2 + n*x*y + y^2 (for n > 1). - Artur Jasinski, Apr 28 2008
a(n)*a(n-1) + 4 = (a(n)-n)^2. This is the case d = 4 in the general (n^2-d)*((n-1)^2-d) + d = (n^2-n-d)^2. - Bruno Berselli, Dec 07 2011
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REFERENCES
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Alain Connes, Noncommutative Geometry, Academic Press, 1994, p. 35.
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LINKS
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FORMULA
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Except for initial term, denominators of energies of hydrogen lines.
a(n) = x, the solution of k = (sqrt(x)+n)/2 and k + (1/k) = n (also valid for a(0) = -4 and a(1) = -3). - Charles L. Hohn, Apr 16 2011
Sum_{n>=3} (-1)^(n+1)/a(n) = 7/48. - Amiram Eldar, Jul 03 2020
Product_{n>=3} (1 - 1/a(n)) = 6*sin(sqrt(5)*Pi)/(sqrt(5)*Pi).
Product_{n>=3} (1 + 1/a(n)) = -4*sqrt(3)*sin(sqrt(3)*Pi)/Pi. (End)
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EXAMPLE
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G.f. = 5*x^3 + 12*x^4 + 21*x^5 + 32*x^6 + 45*x^7 + 60*x^8 + 77*x^9 + 96*x^10 + ...
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {0, 5, 12}, 50] (* G. C. Greubel, Nov 25 2016 *)
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PROG
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CROSSREFS
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a(n), n>=3, second column (used for the Balmer series of the hydrogen atom) of triangle A120070.
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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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