A084157 a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4) with a(0)=0, a(1)=1, a(2)=4, a(3)=22.

0, 1, 4, 22, 112, 556, 2704, 13000, 62080, 295312, 1401664, 6644320, 31472896, 149017792, 705395968, 3338614912, 15800258560, 74772443392, 353840161792, 1674425579008, 7923565146112, 37494981225472, 177428889407488

Offset: 0

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A084156.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-16,0,12).

Crossrefs

Cf. A026150, A083881, A084155, A084156.

Cf. A002605, A003462, A080040.

Programs

Magma

I:=[0,1,4,22]; [n le 4 select I[n] else 8*Self(n-1) -16*Self(n-2) +12*Self(n-4): n in [1..41]]; // G. C. Greubel, Oct 11 2022

Mathematica

LinearRecurrence[{8,-16,0,12},{0,1,4,22},30] (* Harvey P. Dale, Feb 19 2017 *)

SageMath

A083881 = BinaryRecurrenceSequence(6,-6,1,3)
A026150 = BinaryRecurrenceSequence(2,2,1,1)
def A084157(n): return (A083881(n) - A026150(n))/2
[A084157(n) for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

a(n) = (A083881(n) - A026150(n))/2.

a(n) = 8*a(n-1) - 16*a(n-2) + 12*a(n-4).

a(n) = ((3+sqrt(3))^n + (3-sqrt(3))^n - (1+sqrt(3))^n - (1-sqrt(3))^n)/4.

G.f.: x*(1-4*x+6*x^2)/((1-2*x-2*x^2)*(1-6*x+6*x^2)).

E.g.f.: exp(2*x)*sinh(x)*cosh(sqrt(3)*x).

From G. C. Greubel, Oct 11 2022: (Start)

a(2*n) = A003462(n)*A026150(2*n) = A003462(n)*A080040(2*n)/2.

a(2*n+1) = (1/2)*(3^(n+1)*A002605(2*n+1) - A026150(2*n+1)). (End)