A097826 Partial sums of Chebyshev sequence S(n,11) = U(n,11/2) = A004190(n).

1, 12, 132, 1441, 15720, 171480, 1870561, 20404692, 222581052, 2427986881, 26485274640, 288910034160, 3151525101121, 34377866078172, 375005001758772, 4090677153268321, 44622443684192760, 486756203372852040, 5309695793417179681, 57919897524216124452

Offset

0,2

References

R. C. Alperin, A nonlinear recurrence and its relations to Chebyshev polynomials, Fib. Q., 58:2 (2020), 140-142.

Links

Colin Barker, Table of n, a(n) for n = 0..963

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (12,-12,1).

Formula

a(n) = Sum_{k=0..n} S(k, 11), with S(k, 11) = U(k, 11/2) = A004190(k) Chebyshev's polynomials of the second kind.

G.f.: 1/((1-x)*(1 - 11*x + x^2)) = 1/(1 - 12*x + 12*x^2 - x^3).

a(n) = 12*a(n-1) - 12*a(n-2) + a(n-3) with n >= 2, a(-1)=0, a(0)=1, a(1)=12.

a(n) = 11*a(n-1) - a(n-2) + 1 with n >= 1, a(-1)=0, a(0)=1.

a(n) = (S(n+1, 11) - S(n, 11) - 1)/9.

a(n) = (2^(-n)*(-13*2^n + (65 - 18*sqrt(13))*(11 - 3*sqrt(13))^n + (11 + 3*sqrt(13))^n*(65 + 18*sqrt(13))))/117. - Colin Barker, Mar 06 2016

Mathematica

LinearRecurrence[{12, -12, 1}, {1, 12, 132}, 30] (* G. C. Greubel, May 24 2019 *)

Prog

(PARI) Vec(1/((1-x)*(1-11*x+x^2)) + O(x^30)) \\ Colin Barker, Jun 15 2015
(Magma) I:=[1, 12, 132]; [n le 3 select I[n] else 12*Self(n-1)-12*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, May 24 2019
(Sage) (1/((1-x)*(1 - 11*x + x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
(GAP) a:=[1, 12, 132];; for n in [4..30] do a[n]:=12*a[n-1]-12*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019

Crossrefs

Cf. A076765, A097784.

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

Sequence in context: A001336 A118475 A190873 * A010580 A010577 A163055

Adjacent sequences: A097823 A097824 A097825 * A097827 A097828 A097829

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 31 2004

Status

approved