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
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).
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
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
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by Robert G. Wilson v, Feb 06 2002
STATUS
approved