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 A056771 #85 Jul 02 2025 16:02:00
%S A056771 1,17,577,19601,665857,22619537,768398401,26102926097,886731088897,
%T A056771 30122754096401,1023286908188737,34761632124320657,
%U A056771 1180872205318713601,40114893348711941777,1362725501650887306817,46292552162781456490001
%N A056771 a(n) = a(-n) = 34*a(n-1) - a(n-2), and a(0)=1, a(1)=17.
%C A056771 The sequence satisfies the Pell equation a(n)^2 - 18 * A202299(n+1)^2 = 1. - _Vincenzo Librandi_, Dec 19 2011
%C A056771 Also numbers n such that n - 1 and 2*n + 2 are squares. - _Arkadiusz Wesolowski_, Mar 15 2015
%C A056771 And they, n - 1 and 2*n + 2, are the squares of A005319 and A003499. - _Michel Marcus_, Mar 15 2015
%C A056771 This sequence {a(n)} gives all the nonnegative integer solutions of the Pell equation a(n)^2 - 32*(3*A091761(n))^2 = +1. - _Wolfdieter Lang_, Mar 09 2019
%H A056771 Seiichi Manyama, <a href="/A056771/b056771.txt">Table of n, a(n) for n = 0..600</a> (terms 0..200 from Vincenzo Librandi)
%H A056771 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A056771 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A056771 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (34,-1).
%F A056771 a(n) = (r^n + 1/r^n)/2 with r = 17 + sqrt(17^2-1).
%F A056771 a(n) = 16*A001110(n) + 1 = A001541(2n) = (4*A001109(n))^2 + 1 = 3*A001109(2n-1) - A001109(2n-2) = A001109(2n) - 3*A001109(2n-1).
%F A056771 a(n) = T(n, 17) = T(2*n, 3) with T(n, x) Chebyshev's polynomials of the first kind. See A053120. T(n, 3)= A001541(n).
%F A056771 G.f.: (1-17*x)/(1-34*x+x^2).
%F A056771 G.f.: (1 - 17*x / (1 - 288*x / (17 - x))). - _Michael Somos_, Apr 05 2019
%F A056771 a(n) = cosh(2n*arcsinh(sqrt(8))). - _Herbert Kociemba_, Apr 24 2008
%F A056771 a(n) = (a^n + b^n)/2 where a = 17 + 12*sqrt(2) and b = 17 - 12*sqrt(2); sqrt(a(n)-1)/4 = A001109(n). - _James R. Buddenhagen_, Dec 09 2011
%F A056771 a(-n) = a(n). - _Michael Somos_, May 28 2014
%F A056771 a(n) = sqrt(1 + 32*9*A091761(n)^2), n >= 0. See one of the Pell comments above. - _Wolfdieter Lang_, Mar 09 2019
%e A056771 G.f. = 1 + 17*x + 577*x^2 + 19601*x^3 + 665857*x^4 + 22619537*x^5 + ...
%t A056771 LinearRecurrence[{34,-1},{1,17},30] (* _Vincenzo Librandi_, Dec 18 2011 *)
%t A056771 a[ n_] := ChebyshevT[ 2 n, 3]; (* _Michael Somos_, May 28 2014 *)
%o A056771 (Sage) [lucas_number2(n,34,1)/2 for n in range(0,15)] # _Zerinvary Lajos_, Jun 27 2008
%o A056771 (Magma) I:=[1, 17]; [n le 2 select I[n] else 34*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Dec 18 2011
%o A056771 (Maxima) makelist(expand(((17+sqrt(288))^n+(17-sqrt(288))^n))/2, n, 0, 15); /* _Vincenzo Librandi_, Dec 18 2011 */
%o A056771 (PARI) {a(n) = polchebyshev( n, 1, 17)}; /* Michael Somos, Apr 05 2019 */
%Y A056771 Cf. A001075, A001541, A001091, A001079, A023038, A011943, A001081, A023039, A001085 and note relationship with square triangular number sequences A001110 and A001109. A091761.
%Y A056771 Row 3 of array A188644.
%K A056771 nonn,easy
%O A056771 0,2
%A A056771 _Henry Bottomley_, Aug 16 2000
%E A056771 More terms from _James Sellers_, Sep 07 2000
%E A056771 Chebyshev comments from _Wolfdieter Lang_, Nov 29 2002