A189836 a(n) = n^2 + 11.
11, 12, 15, 20, 27, 36, 47, 60, 75, 92, 111, 132, 155, 180, 207, 236, 267, 300, 335, 372, 411, 452, 495, 540, 587, 636, 687, 740, 795, 852, 911, 972, 1035, 1100, 1167, 1236, 1307, 1380, 1455, 1532, 1611, 1692, 1775, 1860, 1947
Offset: 0
Links
- G. C. Greubel, 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 + 11: n in [0..50]]; // G. C. Greubel, Jan 13 2018
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Mathematica
Table[n^2+11,{n,0,100}] LinearRecurrence[{3,-3,1},{11,12,15},60] (* Harvey P. Dale, Aug 24 2020 *) -
PARI
a(n)=n^2+11 \\ Charles R Greathouse IV, Jun 17 2017
Formula
From G. C. Greubel, Jan 13 2018: (Start)
G.f.: (11 - 21*x + 12*x^2)/(1 - x)^3.
E.g.f.: (11 + x + x^2)*exp(x). (End)
From Amiram Eldar, Nov 02 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(11)*Pi*coth(sqrt(11)*Pi))/22.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(11)*Pi*cosech(sqrt(11)*Pi))/22. (End)
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = sqrt(10/11)*sinh(sqrt(10)*Pi)/sinh(sqrt(11)*Pi).
Product_{n>=0} (1 + 1/a(n)) = 2*sqrt(3/11)*sinh(2*sqrt(3)*Pi)/sinh(sqrt(11)*Pi). (End)