A255848 a(n) = 2*n^2 + 18.

18, 20, 26, 36, 50, 68, 90, 116, 146, 180, 218, 260, 306, 356, 410, 468, 530, 596, 666, 740, 818, 900, 986, 1076, 1170, 1268, 1370, 1476, 1586, 1700, 1818, 1940, 2066, 2196, 2330, 2468, 2610, 2756, 2906, 3060, 3218, 3380, 3546, 3716, 3890, 4068, 4250, 4436

Offset: 0

Author

Avi Friedlich, Mar 08 2015

Keywords

• nonn
• easy

Comments

For n>3, the sequence gives the 6th diagonal of triangle in A055096.

Also, this is the case k=9 of the form (n + sqrt(k))^2 + (n - sqrt(k))^2. It is noted that a(n)*n = (n + sqrt(3))^3 + (n - sqrt(3))^3.

Equivalently, numbers m such that 2*m-36 is a square.

Links

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

Crossrefs

Cf. A016825 (first differences), A055096, A189834.

Subsequence of A047463.

Cf. similar sequences listed in A255843.

Programs

Magma

[2*n^2+18: n in [0..50]]; // Vincenzo Librandi, Mar 08 2015

Mathematica

f[n_] := 2 n^2 + 18; Array[f, 50, 0] (* Robert G. Wilson v, Mar 08 2015 *)
CoefficientList[Series[(18 - 34 x + 20 x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Mar 08 2015 *)
LinearRecurrence[{3,-3,1},{18,20,26},50] (* Harvey P. Dale, Aug 20 2021 *)

PARI

vector(50, n, 2*n^2+18) \\ Derek Orr, Mar 09 2015

Sage

[2*n^2+18 for n in (0..50)] # Bruno Berselli, Mar 11 2015

Formula

a(n) = 2*A189834(n).

From Vincenzo Librandi, Mar 08 2015: (Start)

G.f.: 2*(9 - 17*x + 10*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 + 3*Pi*coth(3*Pi))/36.

Sum_{n>=0} (-1)^n/a(n) = (1 + 3*Pi*cosech(3*Pi))/36. (End)

E.g.f.: 2*exp(x)*(9 + x + x^2). - Elmo R. Oliveira, Jan 25 2025

Extensions

Edited by Bruno Berselli, Mar 11 2015