A114964 a(n) = n^2 + 30.

30, 31, 34, 39, 46, 55, 66, 79, 94, 111, 130, 151, 174, 199, 226, 255, 286, 319, 354, 391, 430, 471, 514, 559, 606, 655, 706, 759, 814, 871, 930, 991, 1054, 1119, 1186, 1255, 1326, 1399, 1474, 1551, 1630, 1711, 1794, 1879, 1966, 2055, 2146, 2239, 2334, 2431, 2530

Offset: 0

Author

Cino Hilliard, Feb 21 2006

Keywords

• nonn
• easy

Comments

x^2 + 30 != y^n for all x,y and n > 1, so this is a subsequence of A007916.

From Bruno Berselli, May 12 2014: (Start)

This is the case k=5 of the identity n^2 + k*(k+1) = (Sum_{i=-k..k} (n+i)^3)/((2*k+1)*n).

Similar sequences: A059100 (k=1), A114949 (k=2), A241748 (k=3), A241850 (k=4). (End)

The old name of this sequence was: Numbers of the form x^2 + 30. Also numbers that are not a perfect power.

Examples

			11*4*a(4) = (-1)^3 + 0^3 + 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = 2024. - _Bruno Berselli_, May 12 2014
		

Links

• Shawn A. Broyles, Table of n, a(n) for n = 0..1000
• 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. A007916, A059100, A114949, A241748, A241850.

Programs

Mathematica

Range[0,60]^2+30 (* Harvey P. Dale, Oct 17 2022 *)

PARI

g(n,p) = for(x=0,n,y=x^2+p;print1(y","));

PARI

a(n) = n^2 + 30; \\ Altug Alkan, Apr 30 2018

Formula

From Amiram Eldar, Nov 04 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(30)*Pi*coth(sqrt(30)*Pi))/60.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(30)*Pi*cosech(sqrt(30)*Pi))/60. (End)

From Elmo R. Oliveira, Dec 30 2024: (Start)

G.f.: (30 - 59*x + 31*x^2)/(1 - x)^3.

E.g.f.: (30 + x + x^2)*exp(x).

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

Extensions

New name from Shawn A. Broyles and Altug Alkan, Apr 30 2018