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 A028557 a(n) = n*(n+5). 33
 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 LINKS Shawn A. Broyles, Table of n, a(n) for n = 0..1000 Patrick De Geest, Palindromic Quasipronics of the form n(n+x). Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014. Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42. 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 Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/5 - 47/300. - Amiram Eldar, Jan 15 2021 From Amiram Eldar, Feb 12 2024: (Start) Product_{n>=1} (1 - 1/a(n)) = -24*cos(sqrt(29)*Pi/2)/(7*Pi). Product_{n>=1} (1 + 1/a(n)) = 8*cos(sqrt(21)*Pi/2)/Pi. (End) MATHEMATICA 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. A002522, A055998, A104675. 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 June 12 11:18 EDT 2024. Contains 373331 sequences. (Running on oeis4.)