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%I #24 Sep 08 2022 08:45:07
%S 2,13,76,443,2582,15049,87712,511223,2979626,17366533,101219572,
%T 589950899,3438485822,20040964033,116807298376,680802826223,
%U 3968009658962,23127255127549,134795521106332,785645871510443,4579079707956326,26688832376227513,155553914549408752
%N Bisection (odd part) of Chebyshev sequence with Diophantine property.
%C -8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n) = A077239(n).
%C The even part is A054488(n) with Diophantine companion A077240(n).
%H Colin Barker, <a href="/A077413/b077413.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 (6,-1).
%F a(n) = 6*a(n-1) - a(n-2), a(-1)=-1, a(0)=2.
%F a(n) = 2*S(n, 6)+S(n-1, 6), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 6) = A001109(n+1).
%F G.f.: (2+x)/(1-6*x+x^2).
%F a(n) = (((3-2*sqrt(2))^n*(-7+4*sqrt(2))+(3+2*sqrt(2))^n*(7+4*sqrt(2))))/(4*sqrt(2)). - _Colin Barker_, Oct 12 2015
%e 8*a(1)^2 + 17 = 8*13^2+17 = 1369 = 37^2 = A077239(1)^2.
%t LinearRecurrence[{6,-1}, {2,13}, 30] (* or *) CoefficientList[Series[ (2+x)/(1-6*x+x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Jan 18 2018 *)
%o (PARI) Vec((2+x)/(1-6*x+x^2) + O(x^30)) \\ _Colin Barker_, Jun 16 2015
%o (Magma) I:=[2,13]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 18 2018
%Y Cf. A077241 (even and odd parts).
%K nonn,easy
%O 0,1
%A _Wolfdieter Lang_, Nov 08 2002
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