A189834 a(n) = n^2 + 9.
9, 10, 13, 18, 25, 34, 45, 58, 73, 90, 109, 130, 153, 178, 205, 234, 265, 298, 333, 370, 409, 450, 493, 538, 585, 634, 685, 738, 793, 850, 909, 970, 1033, 1098, 1165, 1234, 1305, 1378, 1453, 1530, 1609, 1690, 1773, 1858, 1945
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..10000
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Programs
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Magma
[n^2+9: n in [0..50]]; // Vincenzo Librandi, Aug 31 2011
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Mathematica
Table[n^2+9,{n,0,100}] LinearRecurrence[{3,-3,1},{9,10,13},50] (* Harvey P. Dale, Aug 21 2020 *) -
PARI
a(n)=n^2+9 \\ Charles R Greathouse IV, Oct 07 2015
Formula
a(n) = A154533(n+1). - R. J. Mathar, May 16 2011
G.f.: ( -9+17*x-10*x^2 ) / (x-1)^3 . - R. J. Mathar, Aug 31 2011
E.g.f.: (9 + x + x^2)*exp(x). - G. C. Greubel, Jan 13 2018
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + 3*Pi*coth(3*Pi))/18.
Sum_{n>=0} (-1)^n/a(n) = (1 + 3*Pi*cosech(3*Pi))/18. (End)
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = (2/3)*sqrt(2)*sinh(2*sqrt(2)*Pi)/sinh(3*Pi).
Product_{n>=0} (1 + 1/a(n)) = (sqrt(10)/3)*sinh(sqrt(10)*Pi)/sinh(3*Pi). (End)