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A077245 Bisection (even part) of Chebyshev sequence with Diophantine property. 4

%I

%S 1,10,79,622,4897,38554,303535,2389726,18814273,148124458,1166181391,

%T 9181326670,72284431969,569094129082,4480468600687,35274654676414,

%U 277716768810625,2186459495808586,17213959197658063

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

%C 3*b(n)^2 - 5*a(n)^2 = 7, with the companion sequence b(n)= A077246(n).

%C The odd part is A077243(n) with Diophantine companion A077244(n).

%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 (8,-1).

%F a(n)= 8*a(n-1) - a(n-2), a(-1) := -2, a(0)=1.

%F a(n)= S(n, 8)+2*S(n-1, 8), with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310. S(-1, x) := 0 and S(n, 8)= A001090(n+1).

%F G.f.: (1+2*x)/(1-8*x+x^2).

%F a(n)=(1/2)*[4-sqrt(15)]^n-(1/5)*[4-sqrt(15)]^n*sqrt(15)+(1/2)*[4+sqrt(15)]^n+(1/5)*sqrt(15) *[4+sqrt(15)]^n, with n>=0 - _Paolo P. Lava_, Jul 08 2008

%e 5*a(1)^2 + 7 = 5*10^2 + 7 = 507 = 3*13^2 = 3*A077246(1)^2.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 08 2002

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Last modified November 15 08:35 EST 2019. Contains 329144 sequences. (Running on oeis4.)