A255844 a(n) = 2*n^2 + 6.

6, 8, 14, 24, 38, 56, 78, 104, 134, 168, 206, 248, 294, 344, 398, 456, 518, 584, 654, 728, 806, 888, 974, 1064, 1158, 1256, 1358, 1464, 1574, 1688, 1806, 1928, 2054, 2184, 2318, 2456, 2598, 2744, 2894, 3048, 3206, 3368, 3534, 3704, 3878, 4056, 4238, 4424, 4614

Offset: 0

Author

Avi Friedlich, Mar 08 2015

Keywords

• nonn
• easy

Comments

This is the case k=3 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. Also, it is noted that a(n)*n = (n + 1)^3 + (n - 1)^3.

Equivalently, numbers m such that 2*m-12 is a square.

For n = 0..16, 3*a(n)-1 is prime (see A087370); for n = 0..12, 3*a(n)-5 is prime (see A107303).

Links

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

Crossrefs

Cf. A016825 (first differences), A087370, A107303, A114949, A117950.

Cf. A152811: nonnegative numbers of the form 2*m^2-6.

Subsequence of A000378.

Cf. similar sequences listed in A255843.

Programs

Magma

[2*n^2+6: n in [0..50]];

Mathematica

Table[2 n^2 + 6, {n, 0, 50}]

PARI

vector(50, n, n--; 2*n^2+6)

Sage

[2*n^2+6 for n in (0..50)]

Formula

G.f.: 2*(3-5*x+4*x^2)/(1 - x)^3.

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

a(n) = 2*A117950(n).

From Amiram Eldar, Mar 28 2023: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(3)*Pi*coth(sqrt(3)*Pi))/12.

Sum_{n>=0} (-1)^n/a(n) = (1 + (sqrt(3)*Pi)*cosech(sqrt(3)*Pi))/12. (End)

E.g.f.: 2*exp(x)*(3 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Extensions

Corrected and extended by Bruno Berselli, Mar 11 2015