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A098603
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a(n) = n*(n+10).
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20
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0, 11, 24, 39, 56, 75, 96, 119, 144, 171, 200, 231, 264, 299, 336, 375, 416, 459, 504, 551, 600, 651, 704, 759, 816, 875, 936, 999, 1064, 1131, 1200, 1271, 1344, 1419, 1496, 1575, 1656, 1739, 1824, 1911, 2000, 2091, 2184, 2279, 2376, 2475, 2576, 2679, 2784
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OFFSET
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0,2
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COMMENTS
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These are the only positive integer values of t for which the Binet-de Moivre formula for the recurrence b(n) = 10*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(10^2+4*t) is a positive integer since 10^2+4*t = 10^2+4*a(m) = (2*m+10)^2. Thus the characteristic roots are r1=10+m and r2 = -m. - Felix P. Muga II, Mar 28 2014
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LINKS
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Shawn A. Broyles, Table of n, a(n) for n = 0..1000
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Wikipedia, Hydrogen spectral series
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).
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FORMULA
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a(n) = (n+5)^2 - 5^2 = n*(n+10), n>=0.
G.f.: x*(11-9*x)/(1-x)^3.
a(n) = 2*n + a(n-1) + 9, (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
Sum_{n>=1} 1/a(n) = 7381/25200 via sum_{n>=0} 1/((n+x)*(n+y)) = (psi(x)-psi(y))/(x-y). - R. J. Mathar, Jul 14 2012
a(0)=0, a(1)=11, a(2)=24, a(n)=3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Jul 26 2014
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MAPLE
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seq(n*(n+10), n=0..53); # Emeric Deutsch, Mar 11 2005
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MATHEMATICA
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s=0; lst={s}; Do[s+=n++ +11; AppendTo[lst, s], {n, 0, 6!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 17 2008 *)
Table[n(n+10), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 11, 24}, 50] (* Harvey P. Dale, Jul 26 2014 *)
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PROG
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(PARI) a(n)=n*(n+10) \\ Charles R Greathouse IV, Jun 16 2017
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CROSSREFS
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Cf. A098832.
a(n-5), n>=6, fifth column (used for the Pfund series of the hydrogen atom) of triangle A120070.
Sequence in context: A213062 A157756 A061043 * A274620 A053061 A055820
Adjacent sequences: A098600 A098601 A098602 * A098604 A098605 A098606
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KEYWORD
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nonn,easy
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AUTHOR
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Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004
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EXTENSIONS
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More terms from Emeric Deutsch, Mar 11 2005
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STATUS
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approved
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