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A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property. 5

%I

%S 7,37,215,1253,7303,42565,248087,1445957,8427655,49119973,286292183,

%T 1668633125,9725506567,56684406277,330380931095,1925601180293,

%U 11223226150663,65413755723685,381259308191447,2222142093424997,12951593252358535,75487417420726213

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

%C a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077413(n).

%C The even part is A077240(n) with Diophantine companion A054488(n).

%H Colin Barker, <a href="/A077239/b077239.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) := 5, a(0)=7.

%F a(n) = 2*T(n+1, 3)+T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).

%F G.f.: (7-5*x)/(1-6*x+x^2).

%F a(n) = (((3-2*sqrt(2))^n*(-8+7*sqrt(2))+(3+2*sqrt(2))^n*(8+7*sqrt(2))))/(2*sqrt(2)). - _Colin Barker_, Oct 12 2015

%e 37 = a(1) = sqrt(8*A077413(1)^2 +17) = sqrt(8*13^2 + 17)= sqrt(1369) = 37.

%t Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 19}] (* _Jean-Fran├žois Alcover_, Dec 19 2013 *)

%o (PARI) Vec((7-5*x)/(1-6*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 12 2015

%Y Cf. A077242 (even and odd parts).

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Nov 08 2002

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)