A114965 a(n) = n^2 + 34.

34, 35, 38, 43, 50, 59, 70, 83, 98, 115, 134, 155, 178, 203, 230, 259, 290, 323, 358, 395, 434, 475, 518, 563, 610, 659, 710, 763, 818, 875, 934, 995, 1058, 1123, 1190, 1259, 1330, 1403, 1478, 1555, 1634, 1715, 1798, 1883, 1970, 2059, 2150, 2243, 2338, 2435

Offset: 0

Author

Cino Hilliard, Feb 21 2006

Keywords

• easy
• nonn

Comments

Conjecture: n^2 + 34 != 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. A114948, A114962, A114963, A114964, A241850.

Programs

Mathematica

34+Range[50]^2  (* Harvey P. Dale, Jan 28 2011 *)

PARI

a(n)=n^2+34

Formula

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

G.f.: (34 - 67*x + 35*x^2)/(1 - x)^3.

E.g.f.: (34 + 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) = 34 prepended by Elmo R. Oliveira, Jan 26 2025