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A028557 n(n+5). 21
0, 6, 14, 24, 36, 50, 66, 84, 104, 126, 150, 176, 204, 234, 266, 300, 336, 374, 414, 456, 500, 546, 594, 644, 696, 750, 806, 864, 924, 986, 1050, 1116, 1184, 1254, 1326, 1400, 1476, 1554, 1634, 1716, 1800, 1886, 1974, 2064, 2156 (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)=5*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(5^2+4*t) is an integer since 5^2+4*t=5^2+4*a(m)=(2*m+5)^2. Thus the characteristic roots are r1=m+5 and r2=-m. - Felix P. Muga II, Mar 27 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..44.

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

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

FORMULA

a(n) = 2*A055998(n).

a(n) = 2*n+a(n-1)+4. - Vincenzo Librandi, Aug 05 2010

sum_{n>=1} 1/a(n) = 137/300 = 0.4566666... - R. J. Mathar, Mar 22 2011

MAPLE

with(combinat): seq(fibonacci(3, n)+n-7, n=2..46); # Zerinvary Lajos, Jun 07 2008

with(finance): seq(add(cashflows([2, k, k], 0), k=2..n), n=1..51); # Zerinvary Lajos, Jun 22 2008

MATHEMATICA

s=0; lst={s}; Do[s+=n++ +6; AppendTo[lst, s], {n, 0, 6!, 2}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)

Table[n(n+5), {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)

PROG

(PARI) a(n)=n*(n+5) \\ Charles R Greathouse IV, Jun 17 2017

CROSSREFS

Cf. A104675, A002522.

Sequence in context: A005281 A120345 A107400 * A104675 A083657 A228435

Adjacent sequences:  A028554 A028555 A028556 * A028558 A028559 A028560

KEYWORD

nonn,easy

AUTHOR

Patrick De Geest

STATUS

approved

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Last modified January 22 22:38 EST 2018. Contains 298093 sequences.