cp's OEIS Frontend

A077234 Bisection (odd part) of Chebyshev sequence with Diophantine property.

%I A077234 #32 Jun 18 2025 00:51:31
%S A077234 2,9,34,127,474,1769,6602,24639,91954,343177,1280754,4779839,17838602,
%T A077234 66574569,248459674,927264127,3460596834,12915123209,48199896002,
%U A077234 179884460799,671337947194,2505467327977,9350531364714,34896658130879,130236101158802,486047746504329
%N A077234 Bisection (odd part) of Chebyshev sequence with Diophantine property.
%C A077234 -3*a(n)^2 + b(n)^2 = 13, with the companion sequence b(n) = A077235(n).
%C A077234 The even part is A054491(n) with Diophantine companion A077236(n).
%H A077234 Colin Barker, <a href="/A077234/b077234.txt">Table of n, a(n) for n = 0..1000</a>
%H A077234 Andrej Dujella and László Szalay, <a href="https://arxiv.org/abs/2506.14013">Four squares from three numbers</a>, arXiv:2506.14013 [math.NT], 2025. See p. 2.
%H A077234 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077234 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077234 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F A077234 a(n) = 2*S(n, 4)+S(n-1, 4), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(-1, x) = 0 and S(n, 4) = A001353(n+1).
%F A077234 G.f.: (2+x)/(1-4*x+x^2).
%F A077234 a(n) = 4*a(n-1)-a(n-2) with a(0)=2 and a(1)=9. - _Philippe Deléham_, Nov 16 2008
%F A077234 E.g.f.: exp(2*x)*(6*cosh(sqrt(3)*x) + 5*sqrt(3)*sinh(sqrt(3)*x))/3. - _Stefano Spezia_, Oct 19 2023
%e A077234 3*a(1)^2 + 13 = 3*81+13 = 256 = 16^2 = A077235(1)^2.
%o A077234 (PARI) Vec((2+x)/(1-4*x+x^2) + O(x^50)) \\ _Colin Barker_, Jun 16 2015
%Y A077234 Cf. A001353, A049310, A054491, A077235, A077236, A077237 (even and odd parts).
%K A077234 nonn,easy
%O A077234 0,1
%A A077234 _Wolfdieter Lang_, Nov 08 2002