A084132 a(n) = 4*a(n-1) + 6*a(n-2), a(0)=1, a(1)=2.

1, 2, 14, 68, 356, 1832, 9464, 48848, 252176, 1301792, 6720224, 34691648, 179087936, 924501632, 4772534144, 24637146368, 127183790336, 656558039552, 3389334900224, 17496687838208, 90322760754176, 466271170045952

Offset: 0

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A002535.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,6).

Crossrefs

Cf. A002535, A005667, A201730.

Programs

Magma

[n le 2 select 2^(n-1) else 4*Self(n-1) +6*Self(n-2): n in [1..40]]; // G. C. Greubel, Oct 13 2022

Mathematica

LinearRecurrence[{4,6}, {1,2}, 40] (* G. C. Greubel, Oct 13 2022 *)

SageMath

[lucas_number2(n,4,-6)/2 for n in range(0, 22)] # Zerinvary Lajos, May 14 2009

SageMath

A084132=BinaryRecurrenceSequence(4,6,1,2)
[A084132(n) for n in range(41)] # G. C. Greubel, Oct 13 2022

Formula

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

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

E.g.f.: exp(2*x)*cosh(sqrt(10)*x).

a(n) = Sum_{k=0..n} A201730(n,k)*9^k. - Philippe Deléham, Dec 06 2011

G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k-2)/(x*(5*k+3) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013

a(n) = 2*i^n*6^((n-2)/2)*( 3*ChebyshevU(n, 2/(i*sqrt(6))) + i*sqrt(6)*ChebyshevU(n -1, 2/(i*sqrt(6))) ). - G. C. Greubel, Oct 13 2022