A255842 a(n) = 2*n^2 + 12.

12, 14, 20, 30, 44, 62, 84, 110, 140, 174, 212, 254, 300, 350, 404, 462, 524, 590, 660, 734, 812, 894, 980, 1070, 1164, 1262, 1364, 1470, 1580, 1694, 1812, 1934, 2060, 2190, 2324, 2462, 2604, 2750, 2900, 3054, 3212, 3374, 3540, 3710, 3884, 4062, 4244, 4430

Offset: 0

Author

Avi Friedlich, Mar 08 2015

Keywords

• nonn
• easy

Comments

This is the case k=6 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + sqrt(2))^3 + (n - sqrt(2))^3.

Equivalently, numbers m such that 2*m - 24 is a square.

For n = 0..10, a(n) - 1 is prime (see A092968).

Links

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

Crossrefs

Cf. A016825 (first differences), A092968, A114949.

Subsequence of A047238 and A047406.

Cf. similar sequences listed in A255843.

Programs

Magma

[2*n^2+12: n in [0..50]];

Mathematica

Table[2 n^2 + 12, {n, 0, 50}]

PARI

vector(50, n, n--; 2*n^2+12)

Sage

[2*n^2+12 for n in (0..50)]

Formula

G.f.: 2*(6 - 11*x + 7*x^2)/(1 - x)^3.

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

a(n) = 2*A114949(n).

From Amiram Eldar, Mar 28 2023: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(6)*Pi*coth(sqrt(6)*Pi))/24.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(6)*Pi*cosech(sqrt(6)*Pi))/24. (End)

E.g.f.: 2*exp(x)*(6 + x + x^2). - Elmo R. Oliveira, Jan 24 2025

Extensions

Edited by Bruno Berselli, Mar 11 2015