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%I #32 Sep 08 2022 08:45:07
%S 1,2,8,13,47,76,274,443,1597,2582,9308,15049,54251,87712,316198,
%T 511223,1842937,2979626,10741424,17366533,62605607,101219572,
%U 364892218,589950899,2126747701,3438485822,12395593988,20040964033,72246816227,116807298376
%N Combined Diophantine Chebyshev sequences A054488 and A077413.
%C -8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n)= A077242(n).
%C The number a > 0 belongs to the sequence A077241, if a^2 belongs to the sequence A034856. - _Alzhekeyev Ascar M_, Apr 27 2012
%C Numbers k such that k^2 + 2 is a triangular number (see A214838). - _Alex Ratushnyak_, Mar 07 2013
%H Vincenzo Librandi, <a href="/A077241/b077241.txt">Table of n, a(n) for n = 0..200</a>
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F a(2k) = A054488(k) and a(2k+1)= A077413(k) for k>=0.
%F G.f.: (1+x)*(1+x+x^2)/(1-6*x^2+x^4).
%F a(n) = (-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8. [_Bruno Berselli_, Mar 10 2013]
%e 8*a(2)^2 + 17 = 8*8^2+17 = 529 = 23^2 = A077242(2)^2.
%t LinearRecurrence[{0, 6, 0, -1}, {1, 2, 8, 13}, 30] (* _Bruno Berselli_, Mar 10 2013 *)
%t CoefficientList[Series[(1 + x) (1 + x + x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 18 2014 *)
%o (Maxima) makelist(expand((-1)^n*((4-5*sqrt(2))*(1-(-1)^n*sqrt(2))^(2*floor((n+1)/2))+(4+5*sqrt(2))*(1+(-1)^n*sqrt(2))^(2*floor((n+1)/2)))/8), n, 0, 30); /* _Bruno Berselli_, Mar 10 2013 */
%o (Magma) I:=[1,2,8,13]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Feb 18 2014
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 08 2002
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