A318610 a(1) = 0, a(2) = 4, a(3) = 12; for n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + 9*a(n-3).

0, 4, 12, 24, 72, 252, 756, 2160, 6480, 19764, 59292, 176904, 530712, 1595052, 4785156, 14346720, 43040160, 129146724, 387440172, 1162241784, 3486725352, 10460412252, 31381236756, 94143001680, 282429005040, 847289140884, 2541867422652, 7625595890664, 22876787671992

Offset

1,2

Comments

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

Links

Jianing Song, Table of n, a(n) for n = 1..500

Index entries for linear recurrences with constant coefficients, signature (3,-3,9).

Formula

a(n) = last 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.: 4*x^2/((1 - 3x)*(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*A318609(n-1).

a(n) = A318609(n) for even n and 2*3^(n-1) - A318609(n) for odd n.

a(n) + A101990(n) + A318609(n) = 3^n.

Example

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

Mathematica

LinearRecurrence[{3, -3, 9}, {0, 4, 12}, 30] (* Vincenzo Librandi, Sep 04 2018 *)

Prog

(PARI) concat([0], Vec(4*x^2/((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]))[3]; \\ Michel Marcus, Dec 20 2019
(Magma) I:=[0, 4, 12]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+9*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 04 2018

Crossrefs

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

A318609 gives the number of solutions to Sum_{i=1..n} x_i^2 == 1 (mod 3).

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

Sequence in context: A356550 A216244 A215223 * A296358 A282512 A025543

Adjacent sequences: A318607 A318608 A318609 * A318611 A318612 A318613

Keyword

nonn,easy

Author

Jianing Song, Sep 02 2018

Status

approved