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A028566
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a(n) = n*(n+8).
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23
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0, 9, 20, 33, 48, 65, 84, 105, 128, 153, 180, 209, 240, 273, 308, 345, 384, 425, 468, 513, 560, 609, 660, 713, 768, 825, 884, 945, 1008, 1073, 1140, 1209, 1280, 1353, 1428, 1505, 1584, 1665, 1748, 1833, 1920, 2009, 2100, 2193, 2288, 2385
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OFFSET
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0,2
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COMMENTS
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a(m) where m is a positive integer are the only positive integer values of t for which the Binet-de Moivre Formula of the recurrence b(n) = 8*b(n-1) + t*b(n-2) with b(0) = 0 and b(1) = 1 has a root which is a square. In particular, sqrt(8^2 + 4*t) is a positive integer since 8^2 + 4*t = 8^2 + 4*a(m) = (2*m + 8)^2. Thus, the characteristics roots are r1 = 8 + m and r2 = -m. - Felix P. Muga II, Mar 28 2014
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LINKS
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FORMULA
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a(n) = (n+4)^2 - 4^2 = n*(n+8), n >= 0.
G.f.: x*(9 - 7*x)/(1 - x)^3.
Sum_{n >= 1} 1/a(n) = 761/2240 = 0.3397321... - R. J. Mathar, Mar 22 2011
Sum_{n>=1} (-1)^(n+1)/a(n) = 533/6720. - Amiram Eldar, Jan 15 2021
Product_{n>=1} (1 - 1/a(n)) = 315*sin(sqrt(17)*Pi)/(13*sqrt(17)*Pi).
Product_{n>=1} (1 + 1/a(n)) = -32*sqrt(15)*sin(sqrt(15)*Pi)/(11*Pi). (End)
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MATHEMATICA
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PROG
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(PARI) a(n)=n*(n+8)
(Scala) (0 to 49).map { n: Int => n * n + 8 * n } // Alonso del Arte, Nov 10 2019
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CROSSREFS
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a(n - 4), n >= 5, fourth column (used for the Brackett 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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