A189834 - OEIS

%I #38 Feb 12 2024 02:28:53

%S 9,10,13,18,25,34,45,58,73,90,109,130,153,178,205,234,265,298,333,370,

%T 409,450,493,538,585,634,685,738,793,850,909,970,1033,1098,1165,1234,

%U 1305,1378,1453,1530,1609,1690,1773,1858,1945

%N a(n) = n^2 + 9.

%H Vincenzo Librandi, <a href="/A189834/b189834.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = A154533(n+1). - _R. J. Mathar_, May 16 2011

%F G.f.: ( -9+17*x-10*x^2 ) / (x-1)^3 . - _R. J. Mathar_, Aug 31 2011

%F E.g.f.: (9 + x + x^2)*exp(x). - _G. C. Greubel_, Jan 13 2018

%F From _Amiram Eldar_, Nov 02 2020: (Start)

%F Sum_{n>=0} 1/a(n) = (1 + 3*Pi*coth(3*Pi))/18.

%F Sum_{n>=0} (-1)^n/a(n) = (1 + 3*Pi*cosech(3*Pi))/18. (End)

%F From _Amiram Eldar_, Feb 12 2024: (Start)

%F Product_{n>=0} (1 - 1/a(n)) = (2/3)*sqrt(2)*sinh(2*sqrt(2)*Pi)/sinh(3*Pi).

%F Product_{n>=0} (1 + 1/a(n)) = (sqrt(10)/3)*sinh(sqrt(10)*Pi)/sinh(3*Pi). (End)

%t Table[n^2+9,{n,0,100}]

%t LinearRecurrence[{3,-3,1},{9,10,13},50] (* _Harvey P. Dale_, Aug 21 2020 *)

%o (Magma) [n^2+9: n in [0..50]]; // _Vincenzo Librandi_, Aug 31 2011

%o (PARI) a(n)=n^2+9 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A002522, A059100, A117950, A087475, A154533.

%K nonn,easy

%O 0,1

%A _Vladimir Joseph Stephan Orlovsky_, Apr 28 2011