OFFSET
0,1
COMMENTS
From César Eliud Lozada, Mar 29 2021: (Start)
Numbers a(n) such that sqrt( a(n) + 4*n*sqrt(2) ) = n + 2*sqrt(2). Examples:
For n=1: sqrt( 9 + 4*sqrt(2)) = 1 + 2*sqrt(2),
For n=2: sqrt(12 + 8*sqrt(2)) = 2 + 2*sqrt(2),
For n=3: sqrt(17 + 12*sqrt(2)) = 3 + 2*sqrt(2). (End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..955 from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
From G. C. Greubel, Jan 13 2018: (Start)
G.f.: (8 - 15*x + 9*x^2)/(1 - x)^3.
E.g.f.: (8 + x + x^2)*exp(x). (End)
From Amiram Eldar, Jul 04 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + 2*sqrt(2)*Pi*coth(2*sqrt(2)*Pi))/16.
Sum_{n>=0} (-1)^n/a(n) = (1 + 2*sqrt(2)*Pi*cosech(2*sqrt(2)*Pi))/16. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = (sqrt(7/2)/2)*sinh(sqrt(7)*Pi)/sinh(2*sqrt(2)*Pi).
Product_{n>=0} (1 + 1/a(n)) = (3/(2*sqrt(2)))*sinh(3*Pi)/sinh(2*sqrt(2)*Pi). (End)
MATHEMATICA
Table[n^2+8, {n, 0, 100}]
LinearRecurrence[{3, -3, 1}, {8, 9, 12}, 50] (* Harvey P. Dale, Jun 21 2022 *)
PROG
(Magma) [n^2+8: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011
(PARI) a(n)=n^2+8 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladimir Joseph Stephan Orlovsky, Apr 28 2011
EXTENSIONS
Offset changed from 1 to 0 by Vincenzo Librandi, Apr 29 2011
STATUS
approved