OFFSET
0,3
LINKS
Index entries for linear recurrences with constant coefficients, signature (4,-7,6,-3).
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
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
a(n) = -26*a(n-6) + 27*a(n-12) for all n in Z. - Michael Somos, Jan 18 2023
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 *)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Jan 23 2008
STATUS
approved