A358027 Expansion of g.f.: (1 + x - 2*x^2 + 2*x^4)/((1-x)*(1-3*x^2)).

1, 2, 3, 6, 11, 20, 35, 62, 107, 188, 323, 566, 971, 1700, 2915, 5102, 8747, 15308, 26243, 45926, 78731, 137780, 236195, 413342, 708587, 1240028, 2125763, 3720086, 6377291, 11160260, 19131875, 33480782, 57395627

Offset: 0

Author

G. C. Greubel, Oct 31 2022

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A052993, A062318, A164123, A254006.

Programs

Magma

I:=[3,6,11]; [1,2] cat [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..60]];

Mathematica

LinearRecurrence[{1,3,-3}, {1,2,3,6,11}, 61]

SageMath

def A254006(n): return 3^(n/2)*(1 + (-1)^n)/2
def A358027(n): return (1/3)*( 4*A254006(n) + 7*A254006(n-1) +2*int(n==0) + 2*int(n==1) - 3 )
[A358027(n) for n in (0..60)]

Formula

a(n) = (1/3)*(2*[n=0] + 2*[n=1] - 3 + 4*A254006(n) + 7*A254006(n-1)).

a(n) = a(n-1) - 3*a(n-2) + 3*a(n-3), for n >= 5.

E.g.f.: (1/3)*( 2 + 2*x - 3*exp(x) + 4*cosh(sqrt(3)*x) + (7/sqrt(3))*sinh(sqrt(3)*x) ).

G.f.: (1 +x -2*x^2 +2*x^4)/((1-x)*(1-3*x^2)). - Clark Kimberling, Oct 31 2022