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A077409 Bisection (even part) of Chebyshev sequence with Diophantine property.

%I A077409 #28 Jul 02 2025 19:27:03
%S A077409 7,59,583,5771,57127,565499,5597863,55413131,548533447,5429921339,
%T A077409 53750679943,532076878091,5267018100967,52138104131579,
%U A077409 516114023214823,5109002128016651,50573907256951687,500630070441500219,4955726797158050503,49056637901139004811
%N A077409 Bisection (even part) of Chebyshev sequence with Diophantine property.
%C A077409 a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077251(n).
%C A077409 The odd part is A077250(n) with Diophantine companion A077249(n).
%H A077409 Colin Barker, <a href="/A077409/b077409.txt">Table of n, a(n) for n = 0..1000</a>
%H A077409 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077409 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077409 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (10,-1).
%F A077409 a(n) = 10*a(n-1)- a(n-2), a(-1)=11, a(0)=7.
%F A077409 a(n) = T(n+1, 5)+2*T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5) = A001079(n).
%F A077409 a(n) = sqrt(24*A077251(n)^2 + 25).
%F A077409 G.f.: (7-11*x)/(1-10*x+x^2).
%e A077409 59 = a(1) = sqrt(24*A077251(1)^2 + 25) = sqrt(24*12^2 + 25) = sqrt(3481) = 59.
%t A077409 CoefficientList[Series[(7 - 11 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* _Vladimir Joseph Stephan Orlovsky_, Jun 11 2011 *)
%t A077409 LinearRecurrence[{10,-1}, {7,59}, 30] (* _G. C. Greubel_, Jan 18 2018 *)
%o A077409 (PARI) a(n)=if(n<0,0,subst(poltchebi(n+1)+2*poltchebi(n),x,5))
%o A077409 (PARI) Vec((7-11*x)/(1-10*x+x^2) + O(x^30)) \\ _Colin Barker_, Jun 15 2015
%o A077409 (PARI) a(n)=polchebyshev(n+1,,5)+2*polchebyshev(n,,5) \\ _Charles R Greathouse IV_, Jun 15 2015
%o A077409 (PARI) a(n)=([0,1;-1,10]^n*[7;59])[1,1] \\ _Charles R Greathouse IV_, Jun 15 2015
%o A077409 (Magma) I:=[7,59]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 18 2018
%K A077409 nonn,easy
%O A077409 0,1
%A A077409 _Wolfdieter Lang_, Nov 08 2002