A241850 a(n) = n^2 + 20.

20, 21, 24, 29, 36, 45, 56, 69, 84, 101, 120, 141, 164, 189, 216, 245, 276, 309, 344, 381, 420, 461, 504, 549, 596, 645, 696, 749, 804, 861, 920, 981, 1044, 1109, 1176, 1245, 1316, 1389, 1464, 1541, 1620, 1701, 1784, 1869, 1956, 2045, 2136, 2229, 2324, 2421, 2520

Offset: 0

Author

Vincenzo Librandi, May 01 2014

Keywords

• nonn
• easy

Comments

The only solution for x at the Diophantine equation x^2 + 20 = y^m (with m > 2) is 14: 14^2 + 20 = a(14) = 6^3. - Bruno Berselli, May 01 2014

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, p. 379.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. similar sequences listed in A114962.

Programs

Magma

[n^2+20: n in [0..60]];

Mathematica

Table[n^2 + 20, {n, 0, 60}]

PARI

a(n)=n^2+20 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (20 - 39*x + 21*x^2)/(1 - x)^3.

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

From Amiram Eldar, Nov 03 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(20)*Pi*coth(sqrt(20)*Pi))/40.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(20)*Pi*cosech(sqrt(20)*Pi))/40. (End)

E.g.f.: exp(x)*(20 + x + x^2). - Elmo R. Oliveira, Nov 29 2024