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A077240 Bisection (even part) of Chebyshev sequence with Diophantine property.

%I A077240 #23 Mar 29 2017 13:17:22
%S A077240 5,23,133,775,4517,26327,153445,894343,5212613,30381335,177075397,
%T A077240 1032071047,6015350885,35060034263,204344854693,1191009093895,
%U A077240 6941709708677,40459249158167,235813785240325,1374423462283783,8010726988462373,46689938468490455
%N A077240 Bisection (even part) of Chebyshev sequence with Diophantine property.
%C A077240 a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A054488(n).
%C A077240 The odd part is A077239(n) with Diophantine companion A077413(n).
%H A077240 Colin Barker, <a href="/A077240/b077240.txt">Table of n, a(n) for n = 0..1000</a>
%H A077240 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077240 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077240 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F A077240 a(n) = 6*a(n-1) - a(n-2), a(-1) = 7, a(0) = 5.
%F A077240 a(n) = T(n+1, 3)+2*T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).
%F A077240 G.f.: (5-7*x)/(1-6*x+x^2).
%F A077240 a(n) = (((3-2*sqrt(2))^n*(-4+5*sqrt(2))+(3+2*sqrt(2))^n*(4+5*sqrt(2))))/(2*sqrt(2)). - _Colin Barker_, Oct 12 2015
%e A077240 23 = a(1) = sqrt(8*A054488(1)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.
%t A077240 Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 19}]  (* _Jean-François Alcover_, Dec 19 2013 *)
%t A077240 LinearRecurrence[{6,-1},{5,23},30] (* _Harvey P. Dale_, Mar 29 2017 *)
%o A077240 (PARI) Vec((5-7*x)/(1-6*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 12 2015
%Y A077240 Cf. A077242 (even and odd parts).
%K A077240 nonn,easy
%O A077240 0,1
%A A077240 _Wolfdieter Lang_, Nov 08 2002