A098297 Member r=12 of the family of Chebyshev sequences S_r(n) defined in A092184.

0, 1, 12, 121, 1200, 11881, 117612, 1164241, 11524800, 114083761, 1129312812, 11179044361, 110661130800, 1095432263641, 10843661505612, 107341182792481, 1062568166419200, 10518340481399521, 104120836647576012

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

S. Barbero, U. Cerruti, and N. Murru, On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (11,-11,1).

Formula

a(n) = (T(n, 5)-1)/4 with Chebyshev's polynomials of the first kind evaluated at x=5: T(n, 5) = A001079(n) = ((5 + 2*sqrt(6))^n + (5 - 2*sqrt(6))^n)/2.

a(n) = 10*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.

a(n) = 11*a(n-1) - 11*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=12.

G.f.: x*(1+x)/((1-x)*(1-10*x+x^2)) = x*(1+x)/(1-11*x+11*x^2-x^3) (from the Stephan link, see A092184).

a(n) = A132596(n) / 2. - Peter Bala, Dec 31 2012

Mathematica

LinearRecurrence[{11, -11, 1}, {0, 1, 12}, 30] (* G. C. Greubel, May 24 2019 *)

Prog

(PARI) my(x='x+O('x^30)); concat([0], Vec(x*(1+x)/((1-x)*(1-10*x+x^2)))) \\ G. C. Greubel, May 24 2019
(Magma) I:=[0, 1, 12]; [n le 3 select I[n] else 11*Self(n-1)-11*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, May 24 2019
(Sage) (x*(1+x)/((1-x)*(1-10*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
(GAP) a:=[0, 1, 12];; for n in [4..30] do a[n]:=11*a[n-1]-11*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019

Crossrefs

Cf. A097784, A098296.

Sequence in context: A222634 A018204 A176779 * A037543 A214317 A037487

Adjacent sequences: A098294 A098295 A098296 * A098298 A098299 A098300

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 18 2004

Status

approved