login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A274975 Sum of n-th powers of the three roots of x^3-2*x^2-x+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^3-2*x^2-x+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*x-3)/(x^3-x^2-2*x+1). - Alois P. Heinz, Jul 14 2016

a(0)=3, a(1)=2, a(2)=6; thereafter a(n)=2*a(n-1)+a(n-2)-a(n-3).

a(n) = (2*cos(Pi/7))^(-n) + (-2*cos(2*Pi/7))^(-n) + (-2*cos(4*Pi/7))^(-n).

a(n) = A033304(n-1) 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*x-3)/(x^3-x^2-2*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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 1 15:30 EST 2020. Contains 338844 sequences. (Running on oeis4.)