|
%I #42 Sep 08 2022 08:45:15
%S 0,1,11,100,891,7921,70400,625681,5560731,49420900,439227371,
%T 3903625441,34693401600,308336988961,2740339499051,24354718502500,
%U 216452127023451,1923714424708561,17096977695353600,151949084833473841
%N Member r=11 of the family of Chebyshev sequences S_r(n) defined in A092184.
%H G. C. Greubel, <a href="/A098296/b098296.txt">Table of n, a(n) for n = 0..1000</a>
%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%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 (10,-10,1).
%F 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.
%F a(n) = 9*a(n-1) - a(n-2) + 2, n >= 2, a(0)=0, a(1)=1.
%F a(n) = 10*a(n-1) - 10*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=11.
%F 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).
%t LinearRecurrence[{10,-10,1},{0,1,11},30] (* _Harvey P. Dale_, Jan 27 2012 *)
%o (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
%o (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
%o (Sage) (x*(1+x)/((1-x)*(1-9*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019
%o (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
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Oct 18 2004
|
