A097833 Partial sums of Chebyshev sequence S(n,20)= U(n,10) = A075843(n+1).

1, 21, 420, 8380, 167181, 3335241, 66537640, 1327417560, 26481813561, 528308853661, 10539695259660, 210265596339540, 4194772231531141, 83685179034283281, 1669508808454134480, 33306490990048406320, 664460310992513991921, 13255899728860231432101, 264453534266212114650100

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Index entries for linear recurrences with constant coefficients, signature (21,-21,1).

Index entries for sequences related to Chebyshev polynomials.

Formula

a(n) = Sum_{k=0..n} S(k, 20), with S(k, 20) = U(k, 10) = A075843(k+1) Chebyshev's polynomials of the second kind.

G.f.: 1/((1-x)*(1-20*x+x^2)) = 1/(1-21*x+21*x^2-x^3).

a(n) = 20*a(n-1)-a(n-2)+1, n>=1, a(-1)=0, a(0)=1.

a(n) = (S(n+1, 20) - S(n, 20) - 1)/18.

a(n) = 21*a(n-1)-21*a(n-2)+a(n-3), n>=2, a(-1)=0, a(0)=1, a(1)=21.

a(n) = (((10+3*sqrt(11))^(-n)*(33+10*sqrt(11)-11*(10+3*sqrt(11))^n*(1257+379*sqrt(11)) + (10+3*sqrt(11))^(2*n)*(262680+79201*sqrt(11)))))/(198*(1257+379*sqrt(11))). - Colin Barker, Mar 03 2016

Sum_{n>=0} 1/a(n) = 11 - 3*sqrt(11). - Amiram Eldar, Jan 31 2026

Mathematica

LinearRecurrence[{21, -21, 1}, {1, 21, 420}, 16] (* Ray Chandler, Aug 11 2015 *)

Crossrefs

Cf. A075843.

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

Sequence in context: A106656 A083043 A162807 * A163145 A163503 A163977

Adjacent sequences: A097830 A097831 A097832 * A097834 A097835 A097836

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved