This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A077242 #27 Sep 08 2022 08:45:07
%S A077242 5,7,23,37,133,215,775,1253,4517,7303,26327,42565,153445,248087,
%T A077242 894343,1445957,5212613,8427655,30381335,49119973,177075397,286292183,
%U A077242 1032071047,1668633125,6015350885,9725506567,35060034263,56684406277,204344854693,330380931095
%N A077242 Combined Diophantine Chebyshev sequences A077240 and A077239.
%C A077242 a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077241(n).
%C A077242 Because there is only one primitive Pythagorean triangle with sum of the legs L = 17 (see also A120681), namely (5,12,13), all positive solutions (x(n), y(n)) = (a(n), 2*A077241(n)) of the (generalized) Pell equation x^2 - 2*y^2 = +17 satisfy x(n) < 2*y(n), for n >= 1, only 5 = x(0) > 2*y(0) = 4. The proof runs along the same line as the one given in a comment on the L=7 case in A077443. - _Wolfdieter Lang_, Feb 05 2015
%H A077242 Vincenzo Librandi, <a href="/A077242/b077242.txt">Table of n, a(n) for n = 0..200</a>
%H A077242 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077242 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,0,-1).
%F A077242 a(2*k) = A077240(k) and a(2*k+1) = A077239(k), k>=0.
%F A077242 G.f.: (1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4).
%F A077242 a(n) = 6*a(n-2)-a(n-4) for n>3. - _Vincenzo Librandi_, Feb 18 2014
%F A077242 a(n) = ((6-5*sqrt(2))*(1-sqrt(2))^n - (-1-sqrt(2))^n*(-4+sqrt(2)) + 4*(-1+sqrt(2))^n + sqrt(2)*(-1+sqrt(2))^n + 6*(1+sqrt(2))^n + 5*sqrt(2)*(1+sqrt(2))^n)/4. - _Colin Barker_, Mar 27 2016
%e A077242 23 = a(2) = sqrt(8*A077241(2)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.
%t A077242 A077239 = Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 12}]; A077240 = Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 12}]; Riffle[A077240, A077239] (* _Jean-François Alcover_, Dec 19 2013 *)
%t A077242 CoefficientList[Series[(1 - x) (5 + 12 x + 5 x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 18 2014 *)
%o A077242 (Magma) I:=[5,7,23,37]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Feb 18 2014
%o A077242 (PARI) Vec((1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4) + O(x^50)) \\ _Colin Barker_, Mar 27 2016
%K A077242 nonn,easy
%O A077242 0,1
%A A077242 _Wolfdieter Lang_, Nov 08 2002