A098296 Member r=11 of the family of Chebyshev sequences S_r(n) defined in A092184.
0, 1, 11, 100, 891, 7921, 70400, 625681, 5560731, 49420900, 439227371, 3903625441, 34693401600, 308336988961, 2740339499051, 24354718502500, 216452127023451, 1923714424708561, 17096977695353600, 151949084833473841
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
- 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 (10,-10,1).
Programs
-
GAP
a:=[0,1,11];; for n in [4..30] do a[n]:=10*a[n-1]-10*a[n-2]+ a[n-3]; od; a; # G. C. Greubel, May 24 2019
-
Magma
I:=[0,1,11]; [n le 3 select I[n] else 10*Self(n-1)-10*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, May 24 2019
-
Mathematica
LinearRecurrence[{10,-10,1},{0,1,11},30] (* Harvey P. Dale, Jan 27 2012 *) -
PARI
my(x='x+O('x^30)); concat([0], Vec(x*(1+x)/((1-x)*(1-9*x+x^2)))) \\ G. C. Greubel, May 24 2019 -
Sage
(x*(1+x)/((1-x)*(1-9*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
Formula
a(n) = 2*(T(n, 9/2)-1)/7 with twice Chebyshev's polynomials of the first kind evaluated at x=9/2: 2*T(n, 9/2) = A056918(n) = ((9 + sqrt(77))^n + (9 - sqrt(77))^n)/2^n.
a(n) = 9*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
a(n) = 10*a(n-1) - 10*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=11.
G.f.: x*(1+x)/((1-x)*(1-9*x+x^2)) = x*(1+x)/(1-10*x+10*x^2-x^3) (from the Stephan link, see A092184).