|
%I #27 Jan 01 2023 14:51:33
%S 3,7,147,1519,18179,208887,2427411,28118111,326005379,3778768231,
%T 43803428627,507757907279,5885832996995,68227336438359,
%U 790877309377939,9167686467977919,106269932920844035,1231859155536345287,14279457438806692755,165524527406049126063,1918726204965152746499
%N Sum of n-th powers of the roots of x^3 - 7*x^2 - 49*x - 49.
%C 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.
%C x1 = -sqrt(7)*tan(Pi/7),
%C x2 = -sqrt(7)*tan(2*Pi/7),
%C x3 = -sqrt(7)*tan(4*Pi/7).
%C a(2n) = A108716(n)* (7^n),
%C a(2n+1) = -A215794(n)* 7^(n+1).
%H Colin Barker, <a href="/A275195/b275195.txt">Table of n, a(n) for n = 0..900</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (7,49,49).
%F G.f.: (-49*x^2-14*x+3)/(-49*x^3-49*x^2-7*x+1).
%F a(n) = (-sqrt(7)*tan(Pi/7))^n + (-sqrt(7)*tan(2*Pi/7))^n + (-sqrt(7)*tan(4*Pi/7))^n.
%F a(0)=3, a(1)=7, a(2)=147; thereafter a(n) = 7*a(n-1) + 49*a(n-2) + 49*a(n-3).
%t 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 *)
%t LinearRecurrence[{7,49,49},{3,7,147},30] (* _Harvey P. Dale_, Jan 01 2023 *)
%o (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++)))))
%o /* Call function as follows to print initial 10 terms: */
%o terms(10) \\ _Felix Fröhlich_, Jul 19 2016
%o (PARI) polsym(x^3 - 7*x^2 - 49*x - 49, 30) \\ _Charles R Greathouse IV_, Jul 20 2016
%o (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
%Y Cf. A108716, A215794.
%K nonn,easy
%O 0,1
%A _Kai Wang_, Jul 19 2016
|
