Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #27 Jun 19 2015 11:19:41
%S 2,21,208,2059,20382,201761,1997228,19770519,195707962,1937309101,
%T 19177383048,189836521379,1879187830742,18602041786041,
%U 184141230029668,1822810258510639,18043961355076722,178616803292256581,1768124071567489088,17502623912382634299
%N Bisection (odd part) of Chebyshev sequence with Diophantine property.
%C -24*a(n)^2 + b(n)^2 = 25, with the companion sequence b(n) = A077250(n).
%C The even part is A077251(n) with Diophantine companion A077409(n).
%H Colin Barker, <a href="/A077249/b077249.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) := -1, a(0)=2.
%F a(n) = 2*S(n, 10)+S(n-1, 10), with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 10)= A004189(n+1).
%F G.f.: (2+x)/(1-10*x+x^2).
%e 24*a(1)^2 + 25 = 24*21^2+25 = 10609 = 103^2 = A077250(1)^2.
%t CoefficientList[Series[(z + 2)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 11 2011 *)
%t LinearRecurrence[{10,-1},{2,21},40] (* _Harvey P. Dale_, Apr 08 2012 *)
%o (PARI) a(n)=if(n<0,0,subst(-7*poltchebi(n)+11*poltchebi(n+1),x,5)/24)
%o (PARI) a(n)=2*polchebyshev(n,2,5)+polchebyshev(n-1,2,5) \\ _Charles R Greathouse IV_, Jun 11 2011
%o (PARI) Vec((2+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