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

0,1

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).

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)

Mathematica

Range[0, 50]^2+5 (* or *) LinearRecurrence[{3, -3, 1}, {5, 6, 9}, 60] (* Harvey P. Dale, Aug 04 2020 *)

Prog

(SageMath) [lucas_number1(3, n, -5) for n in range(0, 51)] # Zerinvary Lajos, May 16 2009
(PARI) a(n)=n^2+5 \\ Charles R Greathouse IV, Apr 10 2012

Crossrefs

For numbers n such that n^2 + 5 is prime, see A078402.

Sequence in context: A301658 A286338 A344231 * A328115 A327975 A227611

Adjacent sequences: A117948 A117949 A117950 * A117952 A117953 A117954

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 04 2006

Status

approved