A077235 - OEIS

%I #23 Dec 31 2023 11:27:08

%S 5,16,59,220,821,3064,11435,42676,159269,594400,2218331,8278924,

%T 30897365,115310536,430344779,1606068580,5993929541,22369649584,

%U 83484668795,311569025596,1162791433589,4339596708760,16195595401451,60442784897044,225575544186725

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

%C a(n)^2 - 3*b(n)^2 = 13, with the companion sequence b(n) = A077234(n).

%C The even part is A077236(n) with Diophantine companion A054491(n).

%H Colin Barker, <a href="/A077235/b077235.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 (4,-1)

%F a(n) = 2*T(n+1, 2)+T(n, 2), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 2)= A001075(n).

%F G.f.: (5-4*x)/(1-4*x+x^2).

%F a(n) = 4*a(n-1)-a(n-2) with a(0)=5 and a(1)=16. - _Philippe Deléham_, Nov 16 2008

%e 16 = a(1) = sqrt(3*A077234(1)^2 + 13) = sqrt(3*9^2 + 13)= sqrt(256) = 16.

%o (PARI) Vec((5-4*x)/(1-4*x+x^2) + O(x^100)) \\ _Colin Barker_, Jun 16 2015

%Y Cf. A077238 (even and odd parts).

%K nonn,easy

%O 0,1

%A _Wolfdieter Lang_, Nov 08 2002