cp's OEIS Frontend

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

%I A189833 #39 Feb 05 2024 02:27:29
%S A189833 8,9,12,17,24,33,44,57,72,89,108,129,152,177,204,233,264,297,332,369,
%T A189833 408,449,492,537,584,633,684,737,792,849,908,969,1032,1097,1164,1233,
%U A189833 1304,1377,1452,1529,1608,1689,1772,1857,1944,2033
%N A189833 a(n) = n^2 + 8.
%C A189833 From _César Eliud Lozada_, Mar 29 2021: (Start)
%C A189833 Numbers a(n) such that sqrt( a(n) + 4*n*sqrt(2) ) = n + 2*sqrt(2). Examples:
%C A189833 For n=1:  sqrt( 9 +  4*sqrt(2)) = 1 + 2*sqrt(2),
%C A189833 For n=2:  sqrt(12 +  8*sqrt(2)) = 2 + 2*sqrt(2),
%C A189833 For n=3:  sqrt(17 + 12*sqrt(2)) = 3 + 2*sqrt(2). (End)
%H A189833 G. C. Greubel, <a href="/A189833/b189833.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..955 from Vincenzo Librandi)
%H A189833 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A189833 From _G. C. Greubel_, Jan 13 2018: (Start)
%F A189833 G.f.: (8 - 15*x + 9*x^2)/(1 - x)^3.
%F A189833 E.g.f.: (8 + x + x^2)*exp(x). (End)
%F A189833 From _Amiram Eldar_, Jul 04 2020: (Start)
%F A189833 Sum_{n>=0} 1/a(n) = (1 + 2*sqrt(2)*Pi*coth(2*sqrt(2)*Pi))/16.
%F A189833 Sum_{n>=0} (-1)^n/a(n) = (1 + 2*sqrt(2)*Pi*cosech(2*sqrt(2)*Pi))/16. (End)
%F A189833 From _Amiram Eldar_, Feb 05 2024: (Start)
%F A189833 Product_{n>=0} (1 - 1/a(n)) = (sqrt(7/2)/2)*sinh(sqrt(7)*Pi)/sinh(2*sqrt(2)*Pi).
%F A189833 Product_{n>=0} (1 + 1/a(n)) = (3/(2*sqrt(2)))*sinh(3*Pi)/sinh(2*sqrt(2)*Pi). (End)
%t A189833 Table[n^2+8,{n,0,100}]
%t A189833 LinearRecurrence[{3,-3,1},{8,9,12},50] (* _Harvey P. Dale_, Jun 21 2022 *)
%o A189833 (Magma) [n^2+8: n in [0..50]]; // _Vincenzo Librandi_, Apr 29 2011
%o A189833 (PARI) a(n)=n^2+8 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A189833 Cf. A002522, A059100, A117950, A087475.
%K A189833 nonn,easy
%O A189833 0,1
%A A189833 _Vladimir Joseph Stephan Orlovsky_, Apr 28 2011
%E A189833 Offset changed from 1 to 0 by _Vincenzo Librandi_, Apr 29 2011