A114948 a(n) = n^2 + 10.

10, 11, 14, 19, 26, 35, 46, 59, 74, 91, 110, 131, 154, 179, 206, 235, 266, 299, 334, 371, 410, 451, 494, 539, 586, 635, 686, 739, 794, 851, 910, 971, 1034, 1099, 1166, 1235, 1306, 1379, 1454, 1531, 1610, 1691, 1774, 1859, 1946, 2035, 2126, 2219, 2314, 2411, 2510

Offset: 0

Author

Cino Hilliard, Feb 21 2006

Keywords

• easy
• nonn

Comments

Conjecture: n^2 + 10 != x^k for all n,x, and k > 1.

The conjecture is true: See Cohn. - James Rayman, Feb 14 2023

Links

• J. H. E. Cohn, The diophantine equation x^2 + C = y^n, Acta Arithmetica LXV.4 (1993).
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A114962, A114963, A114964, A114965, A241850.

Programs

Mathematica

a[n_]:=n^2+10; a[Range[200]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011*)

Formula

From Amiram Eldar, Nov 02 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(10)*Pi*coth(sqrt(10)*Pi))/20.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(10)*Pi*cosech(sqrt(10)*Pi))/20. (End)

From Amiram Eldar, Feb 12 2024: (Start)

Product_{n>=0} (1 - 1/a(n)) = (3/sqrt(10))*sinh(3*Pi)/sinh(sqrt(10)*Pi).

Product_{n>=0} (1 + 1/a(n)) = sqrt(11/10)*sinh(sqrt(11)*Pi)/sinh(sqrt(10)*Pi). (End)

From Elmo R. Oliveira, Jan 25 2025: (Start)

G.f.: (10 - 19*x + 11*x^2)/(1 - x)^3.

E.g.f.: (10 + x + x^2)*exp(x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Extensions

Edited by Charles R Greathouse IV, Aug 09 2010

a(0) = 10 prepended by Elmo R. Oliveira, Jan 26 2025