A255845 a(n) = 2*n^2 + 10.

10, 12, 18, 28, 42, 60, 82, 108, 138, 172, 210, 252, 298, 348, 402, 460, 522, 588, 658, 732, 810, 892, 978, 1068, 1162, 1260, 1362, 1468, 1578, 1692, 1810, 1932, 2058, 2188, 2322, 2460, 2602, 2748, 2898, 3052, 3210, 3372, 3538, 3708, 3882, 4060, 4242, 4428

Offset: 0

Author

Avi Friedlich, Mar 08 2015

Keywords

• nonn
• easy

Comments

This is the case k=5 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 20 is a square.

Links

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

Crossrefs

Cf. A016825 (first differences), A117951.

Subsequence of A047463.

Cf. similar sequences listed in A255843.

Programs

Magma

[2*n^2+10: n in [0..50]]; // Vincenzo Librandi, Mar 08 2015

Mathematica

Table[2 n^2 + 10, {n, 0, 50}]

PARI

a(n)=2*n^2+10 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 2*A117951(n).

From Vincenzo Librandi, Mar 08 2015: (Start)

G.f.: 2*(5 - 9*x + 6*x^2)/(1 - x)^3.

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

From Amiram Eldar, Mar 28 2023: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(5)*Pi*coth(sqrt(5)*Pi))/20.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(5)*Pi*cosech(sqrt(5)*Pi))/20. (End)

E.g.f.: 2*exp(x)*(5 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Extensions

Edited by Bruno Berselli, Mar 13 2015