

A274975


Sum of nth powers of the three roots of x^32*x^2x+1.


9



3, 2, 6, 11, 26, 57, 129, 289, 650, 1460, 3281, 7372, 16565, 37221, 83635, 187926, 422266, 948823, 2131986, 4790529, 10764221, 24186985, 54347662, 122118088, 274396853, 616564132, 1385407029, 3112981337, 6994805571, 15717185450, 35316195134, 79354770147, 178308549978
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,1


COMMENTS

a(n) is x1^n + x2^n + x3^n, where x1, x2, x3 are the roots of the polynomial x^32*x^2x+1.
x1 = 1/(2*cos(Pi/7)),
x2 = 1/(2*cos(2*Pi/7)),
x3 = 1/(2*cos(4*Pi/7)).


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Index entries for linear recurrences with constant coefficients, signature (2,1,1).


FORMULA

G.f.: (x^2+4*x3)/(x^3x^22*x+1).  Alois P. Heinz, Jul 14 2016
a(0)=3, a(1)=2, a(2)=6; thereafter a(n)=2*a(n1)+a(n2)a(n3).
a(n) = (2*cos(Pi/7))^(n) + (2*cos(2*Pi/7))^(n) + (2*cos(4*Pi/7))^(n).
a(n) = A033304(n1) for n>0.


MATHEMATICA

CoefficientList[Series[(x^2 + 4 x  3)/(x^3  x^2  2 x + 1), {x, 0, 32}], x] (* Michael De Vlieger, Jul 14 2016 *)


PROG

(PARI) Vec((x^2+4*x3)/(x^3x^22*x+1) + O(x^50)) \\ Colin Barker, Aug 02 2016


CROSSREFS

Cf. A096975.
3 followed by terms of A033304.
Sequence in context: A109876 A108284 A095011 * A188621 A175182 A291221
Adjacent sequences: A274972 A274973 A274974 * A274976 A274977 A274978


KEYWORD

nonn,easy


AUTHOR

Kai Wang, Jul 14 2016


STATUS

approved



