A345031 a(n) = 6*a(n-1) - 7*a(n-2) - 2*a(n-3) for n >= 3, with a(0) = a(1) = 0, a(2) = 1.

0, 0, 1, 6, 29, 130, 565, 2422, 10317, 43818, 185845, 787710, 3337709, 14140594, 59904181, 253765510, 1074982605, 4553728698, 19289962933, 81713711502, 346145071085, 1466294520130, 6211324200181, 26311593418006, 111457702066509, 472142410072650

Offset: 0

Author

Jianing Song, Jun 06 2021

Keywords

• nonn
• easy

Comments

a(n) is the lower-left entry of {{6, -7, -2}, {1, 0, 0}, {0, 1, 0}}^n.

2p = A100484(k) divides a(p) for odd prime p = prime(k).

a(n) and n have the opposite parity for n >= 1. - Jianing Song, Jun 09 2021

Links

• Jianing Song, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (6,-7,-2).

Crossrefs

Cf. A343575.

Programs

PARI

a(n) = my(M = [6, -7, -2; 1, 0, 0; 0, 1, 0]); (M^n)[3,1]

Formula

a(n) = (((1 + sqrt(5))/2)^(3n) + ((1 - sqrt(5))/2)^(3n) - 2^(n+1))/10.

Relation with A343575:

For even n, A343575(n) = 10*(a(n) mod (2*n)) - 1;

For odd n, A343575(n) = 10*(a(n) mod (2*n)).

O.g.f.: x^2/((1 - 2*x)*(1 - 4*x - x^2)).

E.g.f.: (exp((2 + sqrt(5))*x) + exp((2 - sqrt(5))*x) - 2*exp(2*x))/10.