A134581 a(n) = 4*a(n-1) - 7*a(n-2) + 6*a(n-3) - 3*a(n-4), starting with 0, 1, 2, 3.

0, 1, 2, 3, 4, 4, 0, -13, -40, -81, -122, -122, 0, 365, 1094, 2187, 3280, 3280, 0, -9841, -29524, -59049, -88574, -88574, 0, 265721, 797162, 1594323, 2391484, 2391484, 0, -7174453, -21523360, -43046721, -64570082, -64570082, 0

Offset: 0

Author

Paul Curtz, Jan 23 2008

Keywords

• sign
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).

Programs

Mathematica

LinearRecurrence[{4, -7, 6, -3}, {0, 1, 2, 3}, 50] (* Harvey P. Dale, Dec 06 2013 *)
a[ n_] := Nest[# + RotateRight @ #&, {0, -1, 0, 0, 0, 1}, n][[1]]; (* Michael Somos, Jan 18 2023 *)

Formula

G.f.: x*(1-2*x+2*x^2)/((1-x+x^2)*(1-3*x+3*x^2)). - Jaume Oliver Lafont, Aug 30 2009

a(n) = A140343(n+3) - 2*A140343(n+2) + 2*A140343(n+1). - R. J. Mathar, Nov 21 2012

From Peter Bala, Jul 24 2017: (Start)

a(6*n) = 0;

a(6*n+1) = ((-1)^n*3^(3*n) + 1)/2;

a(6*n+2) = ((-1)^n*3^(3*n+1) + 1)/2;

a(6*n+3) = (-1)^n*3^(3*n+1);

a(6*n+4) = a(6*n+5) = ((-1)^n*3^(3*n+2) - 1)/2.

The o.g.f. A(x) satisfies (1 - x)*A(x) = x*A(1 - x).

Logarithmic g.f.: (1/sqrt(3))*arctan(sqrt(3)*x*(1 - x)/(1 - 2*x)) = Sum_{n >= 1} a(n)*x^n/n.

Sum_{n >= 1} a(n)/(n*2^n) = Pi/(2*sqrt(3)). (End)

a(n) = (3^(n/2) * sin(Pi*n/6) + sin(Pi*n/3)) / sqrt(3). - Peter Luschny, Jul 24 2017

2*a(n) = A010892(n-1) + A057083(n-1). - R. J. Mathar, Oct 03 2021

a(n) = -26*a(n-6) + 27*a(n-12) for all n in Z. - Michael Somos, Jan 18 2023