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 A077413 #27 Jul 31 2024 11:43:27
%S A077413 2,13,76,443,2582,15049,87712,511223,2979626,17366533,101219572,
%T A077413 589950899,3438485822,20040964033,116807298376,680802826223,
%U A077413 3968009658962,23127255127549,134795521106332,785645871510443,4579079707956326,26688832376227513,155553914549408752
%N A077413 Bisection (odd part) of Chebyshev sequence with Diophantine property.
%C A077413 -8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n) = A077239(n).
%C A077413 The even part is A054488(n) with Diophantine companion A077240(n).
%H A077413 Colin Barker, <a href="/A077413/b077413.txt">Table of n, a(n) for n = 0..1000</a>
%H A077413 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077413 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077413 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-1).
%F A077413 a(n) = 6*a(n-1) - a(n-2), a(-1)=-1, a(0)=2.
%F A077413 a(n) = 2*S(n, 6)+S(n-1, 6), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 6) = A001109(n+1).
%F A077413 G.f.: (2+x)/(1-6*x+x^2).
%F A077413 a(n) = (((3-2*sqrt(2))^n*(-7+4*sqrt(2))+(3+2*sqrt(2))^n*(7+4*sqrt(2))))/(4*sqrt(2)). - _Colin Barker_, Oct 12 2015
%F A077413 a(n) = -A054488(-1-n) for all n in Z. - _Michael Somos_, Jul 30 2024
%e A077413 8*a(1)^2 + 17 = 8*13^2+17 = 1369 = 37^2 = A077239(1)^2.
%e A077413 G.f. = 2 + 13*x + 76*x^2 + 443*x^3 + 2582*x^4 + ... - _Michael Somos_, Jul 30 2024
%t A077413 LinearRecurrence[{6,-1}, {2,13}, 30] (* or *) CoefficientList[Series[ (2+x)/(1-6*x+x^2), {x, 0, 50}], x] (* _G. C. Greubel_, Jan 18 2018 *)
%t A077413 a[ n_] := 2*ChebyshevU[n, 3] + ChebyshevU[n-1, 3]; (* _Michael Somos_, Jul 30 2024 *)
%o A077413 (PARI) Vec((2+x)/(1-6*x+x^2) + O(x^30)) \\ _Colin Barker_, Jun 16 2015
%o A077413 (PARI) {a(n) = 2*polchebyshev(n, 2, 3) + polchebyshev(n-1, 2, 3)}; /* _Michael Somos_, Jul 30 2024 */
%o A077413 (PARI) {a(n) = my(w = 3 + quadgen(32)); imag(w^n + 2*w^(n+1))}; /* _Michael Somos_, Jul 30 2024 */
%o A077413 (Magma) I:=[2,13]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 18 2018
%Y A077413 Cf. A077241 (even and odd parts), A001109, A054488.
%K A077413 nonn,easy
%O A077413 0,1
%A A077413 _Wolfdieter Lang_, Nov 08 2002