A255843 a(n) = 2*n^2 + 4.

4, 6, 12, 22, 36, 54, 76, 102, 132, 166, 204, 246, 292, 342, 396, 454, 516, 582, 652, 726, 804, 886, 972, 1062, 1156, 1254, 1356, 1462, 1572, 1686, 1804, 1926, 2052, 2182, 2316, 2454, 2596, 2742, 2892, 3046, 3204, 3366, 3532, 3702, 3876, 4054, 4236, 4422

Offset: 0

Author

Avi Friedlich, Mar 08 2015

Keywords

• nonn
• easy

Comments

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

Equivalently, numbers m such that 2*m - 8 is a square.

Links

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

Crossrefs

Cf. A059100.

Cf. unsigned A147973: numbers of the form 2*m^2-4.

Cf. sequences of the form 2*m^2+2*k: A005893 (k=1), this sequence (k=2), A255844 (k=3), A155966 (k=4), A255845 (k=5), A255842 (k=6), A255846 (k=7), A255847 (k=8), A255848 (k=9).

Programs

Magma

[2*n^2+4: n in [0..50]];

Mathematica

Table[2 n^2 + 4, {n, 0, 50}]

PARI

vector(50, n, n--; 2*n^2+4)

Sage

[2*n^2+4 for n in (0..50)]

Formula

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

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

a(n) = 2*A059100(n).

a(n) = a(n-1) + 4n - 2. - Bob Selcoe, Mar 25 2020

From Amiram Eldar, Mar 28 2023: (Start)

Sum_{n>=0} 1/a(n) = (1 + sqrt(2)*Pi*coth(sqrt(2)*Pi))/8.

Sum_{n>=0} (-1)^n/a(n) = (1 + sqrt(2)*Pi*cosech(sqrt(2)*Pi))/8. (End)

E.g.f.: 2*exp(x)*(2 + x + x^2). - Stefano Spezia, Aug 07 2024

Extensions

Edited by Bruno Berselli, Mar 13 2015