A189833 a(n) = n^2 + 8.

8, 9, 12, 17, 24, 33, 44, 57, 72, 89, 108, 129, 152, 177, 204, 233, 264, 297, 332, 369, 408, 449, 492, 537, 584, 633, 684, 737, 792, 849, 908, 969, 1032, 1097, 1164, 1233, 1304, 1377, 1452, 1529, 1608, 1689, 1772, 1857, 1944, 2033

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Apr 28 2011

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002522, A059100, A117950, A087475.

Programs

Magma

[n^2+8: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

Mathematica

Table[n^2+8,{n,0,100}]
LinearRecurrence[{3,-3,1},{8,9,12},50] (* Harvey P. Dale, Jun 21 2022 *)

PARI

a(n)=n^2+8 \\ Charles R Greathouse IV, Jun 17 2017

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)

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 29 2011