A275195 Sum of n-th powers of the roots of x^3 - 7*x^2 - 49*x - 49.

3, 7, 147, 1519, 18179, 208887, 2427411, 28118111, 326005379, 3778768231, 43803428627, 507757907279, 5885832996995, 68227336438359, 790877309377939, 9167686467977919, 106269932920844035, 1231859155536345287, 14279457438806692755, 165524527406049126063, 1918726204965152746499

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 - 7*x^2 - 49*x -49.

x1 = -sqrt(7)*tan(Pi/7),

x2 = -sqrt(7)*tan(2*Pi/7),

x3 = -sqrt(7)*tan(4*Pi/7).

a(2n) = A108716(n)* (7^n),

a(2n+1) = -A215794(n)* 7^(n+1).

Links

Colin Barker, Table of n, a(n) for n = 0..900

Index entries for linear recurrences with constant coefficients, signature (7,49,49).

Formula

G.f.: (-49*x^2-14*x+3)/(-49*x^3-49*x^2-7*x+1).

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

a(0)=3, a(1)=7, a(2)=147; thereafter a(n) = 7*a(n-1) + 49*a(n-2) + 49*a(n-3).

Mathematica

CoefficientList[Series[(-49 x^2 - 14 x + 3)/(-49 x^3 - 49 x^2 - 7 x + 1), {x, 0, 20}], x] (* Michael De Vlieger, Jul 19 2016 *)
LinearRecurrence[{7, 49, 49}, {3, 7, 147}, 30] (* Harvey P. Dale, Jan 01 2023 *)

Prog

(PARI) terms(n) = my(a=3, b=7, c=147, d=0, i=0); while(i < n, if(i==0, print1(a, ", "); i++, if(i==1, print1(b, ", "); i++, if(i==2, print1(c, ", "); i++, while(i < n, d=7*c + 49*b + 49*a; print1(d, ", "); a=b; b=c; c=d; i++)))))
/* Call function as follows to print initial 10 terms: */
terms(10) \\ Felix Fröhlich, Jul 19 2016
(PARI) polsym(x^3 - 7*x^2 - 49*x - 49, 30) \\ Charles R Greathouse IV, Jul 20 2016
(PARI) Vec((1-7*x)*(3+7*x)/(1-7*x-49*x^2-49*x^3) + O(x^30)) \\ Colin Barker, Jul 23 2016

Crossrefs

Cf. A108716, A215794.

Sequence in context: A006031 A219166 A119958 * A031881 A264931 A227890

Adjacent sequences: A275192 A275193 A275194 * A275196 A275197 A275198

Keyword

nonn,easy

Author

Kai Wang, Jul 19 2016

Status

approved