A255847 a(n) = 2*n^2 + 16.

16, 18, 24, 34, 48, 66, 88, 114, 144, 178, 216, 258, 304, 354, 408, 466, 528, 594, 664, 738, 816, 898, 984, 1074, 1168, 1266, 1368, 1474, 1584, 1698, 1816, 1938, 2064, 2194, 2328, 2466, 2608, 2754, 2904, 3058, 3216, 3378, 3544, 3714, 3888, 4066, 4248, 4434, 4624

Offset: 0

Author

Avi Friedlich, Mar 08 2015

Keywords

• nonn
• easy

Comments

This is the case k=8 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2.

Equivalently, numbers m such that 2*m - 32 is a square.

Links

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

Crossrefs

Cf. A189833.

Subsequence of A047467.

Cf. similar sequences listed in A255843.

Programs

Magma

[2*n^2+16: n in [0..50]];

Mathematica

Table[2 n^2 + 16, {n, 0, 50}]
LinearRecurrence[{3,-3,1},{16,18,24},50] (* Harvey P. Dale, Nov 11 2017 *)

PARI

vector(50, n, n--; 2*n^2+16)

Sage

[2*n^2+16 for n in (0..50)]

Formula

G.f.: 2*(8 - 15*x + 9*x^2)/(1 - x)^3.

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

a(n) = 2*A189833(n).

From Amiram Eldar, Mar 28 2023: (Start)

Sum_{n>=0} 1/a(n) = (1 + 2*sqrt(2)*Pi*coth(2*sqrt(2)*Pi))/32.

Sum_{n>=0} (-1)^n/a(n) = (1 + 2*sqrt(2)*Pi*cosech(2*sqrt(2)*Pi))/32. (End)

E.g.f.: 2*exp(x)*(8 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Extensions

Edited by Bruno Berselli, Mar 13 2015