OFFSET
0,2
COMMENTS
These are the only positive integer values of t for which the Binet-de Moivre formula for the recurrence b(n) = 10*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(10^2+4*t) is a positive integer since 10^2+4*t = 10^2+4*a(m) = (2*m+10)^2. Thus the characteristic roots are r1=10+m and r2 = -m. - Felix P. Muga II, Mar 28 2014
LINKS
Shawn A. Broyles, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Wikipedia, Hydrogen spectral series.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = (n+5)^2 - 5^2 = n*(n+10), n>=0.
G.f.: x*(11-9*x)/(1-x)^3.
a(n) = a(n-1) + 2*n + 9, (with a(0)=0). - Vincenzo Librandi, Nov 17 2010
Sum_{n>=1} 1/a(n) = 7381/25200 via sum_{n>=0} 1/((n+x)*(n+y)) = (psi(x)-psi(y))/(x-y). - R. J. Mathar, Jul 14 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=0, a(1)=11, a(2)=24. - Harvey P. Dale, Jul 26 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = 1627/25200. - Amiram Eldar, Jan 15 2021
E.g.f.: x*(11 + x)*exp(x). - G. C. Greubel, Jul 31 2022
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -18144*sqrt(2/13)*sin(sqrt(26)*Pi)/(935*Pi).
Product_{n>=1} (1 + 1/a(n)) = 126*sqrt(6)*sin(2*sqrt(6)*Pi)/(23*Pi). (End)
MAPLE
seq(n*(n+10), n=0..53); # Emeric Deutsch, Mar 11 2005
MATHEMATICA
Table[n(n+10), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 11, 24}, 50] (* Harvey P. Dale, Jul 26 2014 *)
PROG
(PARI) a(n)=n*(n+10) \\ Charles R Greathouse IV, Jun 16 2017
(Magma) [n*(n+10): n in [0..50]]; // G. C. Greubel, Jul 31 2022
(SageMath) [n*(n+10) for n in (0..50)] # G. C. Greubel, Jul 31 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Eugene McDonnell (eemcd(AT)mac.com), Nov 04 2004
EXTENSIONS
More terms from Emeric Deutsch, Mar 11 2005
STATUS
approved