A114963 a(n) = n^2 + 22.

22, 23, 26, 31, 38, 47, 58, 71, 86, 103, 122, 143, 166, 191, 218, 247, 278, 311, 346, 383, 422, 463, 506, 551, 598, 647, 698, 751, 806, 863, 922, 983, 1046, 1111, 1178, 1247, 1318, 1391, 1466, 1543, 1622, 1703, 1786, 1871, 1958, 2047, 2138, 2231, 2326, 2423, 2522, 2623

Offset: 0

Author

Cino Hilliard, Feb 21 2006

Keywords

• nonn
• easy

Comments

Old name was: "Numbers of the form x^2 + 22".

x^2 + 22 != y^n for all x,y and n > 1.

Links

• Vincenzo Librandi, 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. similar sequences listed in A114962.

Programs

Magma

[n^2+22: n in [0..60]]; // Vincenzo Librandi, Apr 30 2014

Mathematica

Table[n^2 + 22, {n, 0, 60}] (* Vincenzo Librandi, Apr 30 2014 *)

PARI

a(n)=n^2+22 \\ Amiram Eldar, Nov 04 2020

Formula

G.f.: (22 - 43*x + 23*x^2)/(1 - x)^3. - Vincenzo Librandi, Apr 30 2014

From Amiram Eldar, Nov 04 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(22)*Pi*coth(sqrt(22)*Pi))/44.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(22)*Pi*cosech(sqrt(22)*Pi))/44. (End)

From Elmo R. Oliveira, Nov 29 2024: (Start)

E.g.f.: exp(x)*(22 + x + x^2).

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

Extensions

a(0)=22 from Vincenzo Librandi, Apr 30 2014

Definition changed by Bruno Berselli, Mar 13 2015

Offset corrected by Amiram Eldar, Nov 04 2020