A028560 a(n) = n*(n + 6).

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

Author

Patrick De Geest

Keywords

• nonn
• easy

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

Also, numbers k such that k + 9 is a perfect square.

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).

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.

Programs

Haskell

a028560 n = n * (n + 6)  -- Reinhard Zumkeller, Apr 07 2013

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}]

PARI

a(n)=n*(n+6) \\ Charles R Greathouse IV, Sep 24 2015

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)

Extensions

Edited by Robert G. Wilson v, Feb 06 2002