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
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).
Programs
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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
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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
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Mathematica
LinearRecurrence[{11,-11,1}, {0,1,12}, 30] (* G. C. Greubel, May 24 2019 *) -
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 -
Sage
(x*(1+x)/((1-x)*(1-10*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 24 2019
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