A028560 a(n) = n*(n + 6), also numbers j such that 9*(9 + j) is a perfect square.

0, 7, 16, 27, 40, 55, 72, 91, 112, 135, 160, 187, 216, 247, 280, 315, 352, 391, 432, 475, 520, 567, 616, 667, 720, 775, 832, 891, 952, 1015, 1080, 1147, 1216, 1287, 1360, 1435, 1512, 1591, 1672, 1755, 1840, 1927, 2016, 2107, 2200, 2295, 2392

Offset

0,2

Comments

Nonnegative X values of solutions to the equation X + (X + 3)^2 + (X + 6)^3 = Y^2. To prove that X = n^2 + 6n: Y^2 = X + (X + 3)^2 + (X + 6)^3 = X^3 + 19*X^2 + 115X + 225 = (X + 9)*(X^2 + 10X + 25) = (X + 9)*(X + 5)^2 it means: (X + 9) must be a perfect square, so X = k^2 - 9 with k>=3. we can put: k = n + 3, which gives: X = n^2 + 6n and Y = (n + 3)*(n^2 + 6n + 5). - Mohamed Bouhamida, Nov 12 2007

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)=6*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(6^2+4*t) is an integer since 6^2+4*t=6^2+4*a(m)=(2*m+6)^2. Thus, the charcteristic roots are k1=6+m and k2=-m. - Felix P. Muga II, Mar 27 2014

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.

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

Leo Tavares, Illustration: Diamond Pairs.

Wikipedia, Hydrogen spectral series.

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

Formula

a(n) = (n+3)^2 - 3^2 = n*(n+6).

G.f.: x*(7-5*x)/(1-x)^3.

a(n) = 2*n + a(n-1) + 5. - Vincenzo Librandi, Aug 05 2010

Sum_{n>=1} 1/a(n) = 49/120 = 0.4083333... - R. J. Mathar, Mar 22 2011

a(n) = A028884(n) - 1. - Reinhard Zumkeller, Apr 07 2013

E.g.f.: x*(x+7)*exp(x). - G. C. Greubel, Aug 19 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = 37/360. - Amiram Eldar, Nov 04 2020

a(n) = A056220(n+1) - A000290(n-1). - Leo Tavares, Sep 29 2022

From Amiram Eldar, Feb 05 2024: (Start)

Product_{n>=1} (1 - 1/a(n)) = -4*sqrt(10)*sin(sqrt(10)*Pi)/(3*Pi).

Product_{n>=1} (1 + 1/a(n)) = 45*sqrt(2)*sin(2*sqrt(2)*Pi)/(7*Pi). (End)

Maple

A028560:=n->n*(n + 6); seq(A028560(n), n=0..100); # Wesley Ivan Hurt, Mar 27 2014

Mathematica

Table[n(n + 6), {n, 0, 65}]

Prog

(Haskell)
a028560 n = n * (n + 6)  -- Reinhard Zumkeller, Apr 07 2013
(PARI) a(n)=n*(n+6) \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

a(n-3), n>=4, third column (used for the Paschen series of the hydrogen atom) of triangle A120070.

Cf. A005563.

Cf. A056220, A000290.

Sequence in context: A052221 A119461 A326664 * A190530 A345071 A351044

Adjacent sequences: A028557 A028558 A028559 * A028561 A028562 A028563

Keyword

nonn,easy

Author

Patrick De Geest

Extensions

Edited by Robert G. Wilson v, Feb 06 2002

Status

approved