A271996 The crystallogen sequence (a(n) = A018227(n)-4).

6, 14, 32, 50, 82, 114, 164, 214, 286, 358, 456, 554, 682, 810, 972, 1134, 1334, 1534, 1776, 2018, 2306, 2594, 2932, 3270, 3662, 4054, 4504, 4954, 5466, 5978, 6556, 7134, 7782, 8430, 9152, 9874, 10674, 11474, 12356, 13238, 14206, 15174, 16232, 17290, 18442

Offset: 2

Author

Natan Arie Consigli, Jun 18 2016

Keywords

• nonn
• easy

Comments

Terms up to 114 are the atomic numbers of the elements of group 14 in the periodic table. Those elements are also called crystallogens.

Links

• Colin Barker, Table of n, a(n) for n = 2..1000
• Wikipedia, Carbon group.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A173592, A018227.

Programs

Mathematica

Table[(6*(-9+(-1)^n)+(25+3*(-1)^n)*n+12*n^2+2*n^3)/12, {n, 2, 10}] (* or *) LinearRecurrence[{2, 1, -4, 1, 2, -1}, {6, 14, 32, 50, 82, 114}, 50] (* G. C. Greubel, Jun 23 2016 *)

PARI

Vec(2*x^2*(3+x-x^2-2*x^3+x^5)/((1-x)^4*(1+x)^2) + O(x^100)) \\ Colin Barker, Jun 19 2016

Formula

From Colin Barker, Jun 19 2016: (Start)

a(n) = (6*(-9 + (-1)^n) + (25 + 3*(-1)^n)*n + 12*n^2 + 2*n^3)/12.

a(n) = (n^3 + 6*n^2 + 14*n - 24)/6 for n even.

a(n) = (n^3 + 6*n^2 + 11*n - 30)/6 for n odd.

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>7.

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

(End)