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A098603 a(n) = n*(n+10). 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

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).

FORMULA

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

MAPLE

seq(n*(n+10), n=0..53); # Emeric Deutsch, Mar 11 2005

MATHEMATICA

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 *)

PROG

(PARI) a(n)=n*(n+10) \\ Charles R Greathouse IV, Jun 16 2017

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004

EXTENSIONS

More terms from Emeric Deutsch, Mar 11 2005

STATUS

approved

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Last modified January 21 19:42 EST 2019. Contains 319350 sequences. (Running on oeis4.)