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A028566 n(n+8). 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.

LINKS

Table of n, a(n) for n=0..45.

Patrick De Geest, Palindromic Quasipronics of the form n(n+x)

Wikipedia, Hydrogen spectral series

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = (n+4)^2 - 4^2 = n*(n+8), n>=0.

G.f.: x*(9-7*x)/(1-x)^3.

a(n) = 2*n + a(n-1) + 7. - Vincenzo Librandi, Aug 05 2010

sum_{n>=1} 1/a(n) = 761/2240 = 0.3397321.. - R. J. Mathar, Mar 22 2011

MATHEMATICA

Table[n (n+8), {n, 0, 50}] (* Bruno Berselli, Apr 06 2014 *)

PROG

(PARI) a(n)=n*(n+8)

(Sage) [n*(n+8) for n in [0..50]] # Bruno Berselli, Apr 06 2014

CROSSREFS

a(n-4), n>=5, fourth column (used for the Brackett series of the hydrogen atom) of triangle A120070.

Sequence in context: A256383 A017497 A059108 * A147479 A146680 A143704

Adjacent sequences:  A028563 A028564 A028565 * A028567 A028568 A028569

KEYWORD

nonn,easy

AUTHOR

Patrick De Geest

STATUS

approved

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Last modified July 25 15:34 EDT 2017. Contains 289795 sequences.