A098298 - OEIS

%I #35 Sep 08 2022 08:45:15

%S 0,1,13,144,1573,17161,187200,2042041,22275253,242985744,2650567933,

%T 28913261521,315395308800,3440435135281,37529391179293,

%U 409382867836944,4465682155027093,48713120837461081,531378647057044800,5796451996790031721,63229593317633304133

%N Member r=13 of the family of Chebyshev sequences S_r(n) defined in A092184.

%H Colin Barker, <a href="/A098298/b098298.txt">Table of n, a(n) for n = 0..950</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 (12,-12,1).

%F a(n) = 2*(T(n, 11/2) - 1)/9 with twice Chebyshev's polynomials of the first kind evaluated at x=11/2: 2*T(n, 11/2) = A057076(n) = ((11 + sqrt(117))^n + (11 - sqrt(117))^n)/2^n.

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

%F a(n) = 12*a(n-1) - 12*a(n-2) + a(n-3), n >= 3, a(0)=0, a(1)=1, a(2)=13.

%F G.f.: x*(1+x)/((1-x)*(1-11*x+x^2)) = x*(1+x)/(1-12*x+12*x^2-x^3) (from the Stephan link, see A092184).

%t LinearRecurrence[{12,-12,1},{0,1,13},30] (* _Harvey P. Dale_, May 11 2012 *)

%t RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == 11 a[n-1] - a[n-2] + 2}, a, {n, 30}] (* _Vincenzo Librandi_, Mar 06 2016 *)

%o (PARI) concat(0, Vec(x*(1+x)/((1-x)*(1-11*x+x^2)) + O(x^25))) \\ _Colin Barker_, Mar 06 2016

%o (Magma) [n le 2 select n-1 else 11*Self(n-1)- Self(n-2) + 2: n in [1..30]]; // _Vincenzo Librandi_, Mar 06 2016

%o (Sage) (x*(1+x)/((1-x)*(1-11*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019

%o (GAP) a:=[0,1,13];; for n in [4..30] do a[n]:=12*a[n-1]-12*a[n-2]+ a[n-3]; od; a; # _G. C. Greubel_, May 24 2019

%Y Cf. A098296, A098297.

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Oct 18 2004