A056124 a(n) = 3*a(n-1) - a(n-2) + 8 with a(0)=1, a(1)=11.

1, 11, 40, 117, 319, 848, 2233, 5859, 15352, 40205, 105271, 275616, 721585, 1889147, 4945864, 12948453, 33899503, 88750064, 232350697, 608302035, 1592555416, 4169364221, 10915537255, 28577247552, 74816205409

Offset: 0

Author

Barry E. Williams, Jul 07 2000

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Crossrefs

Cf. A000032, A000045, A055850 (first differences).

Programs

GAP

List([0..30], n-> Fibonacci(2*n+2) + 8*Fibonacci(2*n+1) - 8 ); # G. C. Greubel, Jan 19 2020

Magma

[Fibonacci(2*n+2) + 8*Fibonacci(2*n+1) - 8: n in [0..30]]; // G. C. Greubel, Jan 19 2020

Maple

with(combinat); seq( fibonacci(2*n+2) + 8*fibonacci(2*n+1) - 8, n=0..30); # G. C. Greubel, Jan 19 2020

Mathematica

LinearRecurrence[{4,-4,1},{1,11,40},30] (* Harvey P. Dale, Mar 25 2015 *)

PARI

vector(31, n, fibonacci(2*n) +8*fibonacci(2*n-1) -8 ) \\ G. C. Greubel, Jan 19 2020

Sage

[fibonacci(2*n+2) + 8*fibonacci(2*n+1) - 8 for n in (0..30)] # G. C. Greubel, Jan 19 2020

Formula

a(n) = ( 19*(((3+sqrt(5))/2)^n - ((3-sqrt(5))/2)^n) - 9*(((3+sqrt(5))/2)^(n-1) - ((3-sqrt(5))/2)^(n-1)) )/sqrt(5) - 8.

G.f.: (1+7*x)/((1-x)*(1-3*x+x^2)).

a(n) = Fibonacci(2*n+5) + 2*Lucas(2*n) - 8.

From G. C. Greubel, Jan 19 2020: (Start)

a(n) = Fibonacci(2*n+2) + 8*Fibonacci(2*n+1) - 8.

E.g.f.: exp(3*x/2)*( 9*cosh(sqrt(5)*x/2) - (11/sqrt(5))*sinh(sqrt(5)*x/2) ) - 8*exp(x). (End)