A117951 a(n) = n^2 + 5.
5, 6, 9, 14, 21, 30, 41, 54, 69, 86, 105, 126, 149, 174, 201, 230, 261, 294, 329, 366, 405, 446, 489, 534, 581, 630, 681, 734, 789, 846, 905, 966, 1029, 1094, 1161, 1230, 1301, 1374, 1449, 1526, 1605, 1686, 1769, 1854, 1941, 2030, 2121, 2214, 2309, 2406, 2505
Offset: 0
Links
- Ivan Panchenko, Table of n, a(n) for n = 0..1000
- Eric Weisstein's World of Mathematics, Near-Square Prime.
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
Crossrefs
Cf. A078402.
Programs
-
Mathematica
Range[0,50]^2+5 (* or *) LinearRecurrence[{3,-3,1},{5,6,9},60] (* Harvey P. Dale, Aug 04 2020 *) -
PARI
a(n)=n^2+5 \\ Charles R Greathouse IV, Apr 10 2012
-
SageMath
[lucas_number1(3,n,-5) for n in range(0, 51)] # Zerinvary Lajos, May 16 2009
Formula
a(n) = 2*n + a(n-1) - 1 (with a(0)=5). - Vincenzo Librandi, Nov 13 2010
From Colin Barker, Apr 10 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: (5-9*x+6*x^2)/(1-x)^3. (End)
From Amiram Eldar, Jul 13 2020: (Start)
Sum_{n>=0} 1/a(n) = (1 + sqrt(5)*Pi*coth(sqrt(5)*Pi))/10.
Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(5)*Pi*cosech(sqrt(5)*Pi))/10. (End)
From Amiram Eldar, Feb 05 2024: (Start)
Product_{n>=0} (1 - 1/a(n)) = 2*sinh(2*Pi)/(sqrt(5)*sinh(sqrt(5)*Pi)).
Product_{n>=0} (1 + 1/a(n)) = sqrt(6/5)*sinh(sqrt(6)*Pi)/sinh(sqrt(5)*Pi). (End)
E.g.f.: exp(x)*(5 + x + x^2). - Elmo R. Oliveira, Jan 17 2025