A077250 - OEIS

%I #22 Jun 19 2015 11:19:41

%S 11,103,1019,10087,99851,988423,9784379,96855367,958769291,9490837543,

%T 93949606139,930005223847,9206102632331,91131021099463,

%U 902104108362299,8929910062523527,88396996516872971,875040055106206183,8662003554545188859,85744995490345682407

%N Bisection (odd part) of Chebyshev sequence with Diophantine property.

%C a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077249(n).

%C The even part is A077409(n) with Diophantine companion A077251(n).

%H Colin Barker, <a href="/A077250/b077250.txt">Table of n, a(n) for n = 0..1000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).

%F a(n) = 10*a(n-1)- a(n-2), a(-1)=7, a(0)=11.

%F a(n) = 2*T(n+1, 5)+T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5)= A001079(n).

%F a(n) = sqrt(25 + 24*A077249(n)^2).

%F G.f.: (11-7*x)/(1-10*x+x^2).

%e 103 = a(1) = sqrt(24*A077249(1)^2 + 25) = sqrt(24*21^2 + 25) = sqrt(10609) = 103.

%t CoefficientList[Series[(11 - 7 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 11 2011 *)

%o (PARI) a(n)= 2*polchebyshev(n+1,1,5)+polchebyshev(n,1,5) \\ _Charles R Greathouse IV_, Jun 11 2011

%o (PARI) Vec((11-7*x)/(1-10*x+x^2) + O(x^30)) \\ _Colin Barker_, Jun 15 2015

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Nov 08 2002