A318609 a(1) = 2, a(2) = 4, a(3) = 6; for n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + 9*a(n-3).

2, 4, 6, 24, 90, 252, 702, 2160, 6642, 19764, 58806, 176904, 532170, 1595052, 4780782, 14346720, 43053282, 129146724, 387400806, 1162241784, 3486843450, 10460412252, 31380882462, 94143001680, 282430067922, 847289140884, 2541864234006, 7625595890664, 22876797237930

Offset: 1

Author

Jianing Song, Sep 02 2018

Keywords

• nonn
• easy

Comments

a(n) is the number of solutions to Sum_{i=1..n} x_i^2 == 1 (mod 3).

Examples

			a(5) = 90 since M^5 * [1, 0, 0]^T = [81, 90, 72]^T.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..500
• Index entries for linear recurrences with constant coefficients, signature (3,-3,9).

Crossrefs

A101990 gives the number of solutions to Sum_{i=1..n} x_i^2 == 0 (mod 3);

A318610 gives the number of solutions to Sum_{i=1..n} x_i^2 == 2 (mod 3).

Cf. A228921, A228920, A229138, A330607, A330619, A330635.

Programs

Magma

I:=[2,4,6]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+9*Self(n-3): n in [1..30]]; // Jianing Song, Sep 05 2018

Mathematica

LinearRecurrence[{3, -3, 9}, {2, 4, 6}, 30] (* Jianing Song, Sep 05 2018 *)

PARI

Vec(2*x*(1-x)/((1-3*x)*(1+3*x^2)) + O(x^40))

PARI

a(n) = ([1, 0, 2 ; 2, 1, 0 ; 0, 2, 1]^n*mattranspose([1, 0, 0]))[2]; \\ Michel Marcus, Dec 20 2019

Formula

a(n) = middle term in M^n * [1, 0, 0]^T, where M = the 3 X 3 matrix [1, 0, 2 / 2, 1, 0 / 0, 2, 1] and T denotes transpose. [Edited by Petros Hadjicostas, Dec 19 2019]

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

E.g.f.: 1/3*(exp(3*x) + 2*cos(sqrt(3)*x - 2*Pi/3)).

a(n) = 3^(n/2 - 1)*((-i)^n*(-1 + sqrt(3)*i)/2 + i^n*(-1 - sqrt(3)*i)/2 + 3^(n/2)), where i is the imaginary unit.

a(n) = 3^(n/2 - 1)*(2*cos(n*Pi/2 - 2*Pi/3) + 3^(n/2)).

a(n) = 3^(n-1) + (-3)^(n/2-1) for even n and 3^(n-1) + (-3)^((n-1)/2) for odd n.

a(n) = a(n-1) + 2*A101990(n-1).

a(n) = A318610(n) for even n and 2*3^(n-1) - A318610(n) for odd n.

a(n) + A101990(n) + A318610(n) = 3^n.