A098297 - OEIS

%I #33 Sep 08 2022 08:45:15

%S 0,1,12,121,1200,11881,117612,1164241,11524800,114083761,1129312812,

%T 11179044361,110661130800,1095432263641,10843661505612,

%U 107341182792481,1062568166419200,10518340481399521,104120836647576012

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

%H G. C. Greubel, <a href="/A098297/b098297.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Barbero, U. Cerruti, and N. Murru, <a href="http://www.seminariomatematico.polito.it/rendiconti/78-1/BarberoCerrutiMurru.pdf">On polynomial solutions of the Diophantine equation (x + y - 1)^2 = wxy</a>, Rendiconti Sem. Mat. Univ. Pol. Torino (2020) Vol. 78, No. 1, 5-12.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-11,1).

%F 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.

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

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

%F 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).

%F a(n) = A132596(n) / 2. - _Peter Bala_, Dec 31 2012

%t LinearRecurrence[{11,-11,1}, {0,1,12}, 30] (* _G. C. Greubel_, May 24 2019 *)

%o (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

%o (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

%o (Sage) (x*(1+x)/((1-x)*(1-10*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019

%o (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

%Y Cf. A097784, A098296.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Oct 18 2004