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%I #67 Aug 27 2023 10:28:40
%S 1,17,288,4879,82655,1400256,23721697,401868593,6808044384,
%T 115334885935,1953885016511,33100710394752,560758191694273,
%U 9499788548407889,160935647131239840,2726406212682669391,46187969968474139807,782469083251377707328,13255786445304946884769
%N A Chebyshev S-sequence with Diophantine property.
%C a(n) gives the general (positive integer) solution of the Pell equation b^2 - 285*a^2 = +4 with companion sequence b(n)=A078367(n+1), n >= 0.
%C This is the m=19 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..18 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364 and A077412. The m=1..3 (signed) sequences are A049347, A056594, A010892.
%C For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 17's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - _John M. Campbell_, Jul 08 2011
%C For n >= 2, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,16}. - _Milan Janjic_, Jan 23 2015
%H Vincenzo Librandi, <a href="/A078366/b078366.txt">Table of n, a(n) for n = 0..800</a>
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=17, q=-1.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eq.(44), lhs, m=19.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (17,-1).
%F a(n) = 17*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1.
%F a(n) = S(2*n+1, sqrt(19))/sqrt(19) = S(n, 17), where S(n, x) = U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
%F a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (17+sqrt(285))/2 and am = (17-sqrt(285))/2.
%F G.f.: 1/(1-17*x+x^2).
%F a(n) = Sum_{k=0..n} A101950(n,k)*16^k. - _Philippe Deléham_, Feb 10 2012
%F Product {n >= 0} (1 + 1/a(n)) = (1/15)*(15 + sqrt(285)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 1} (1 - 1/a(n)) = (1/34)*(15 + sqrt(285)). - _Peter Bala_, Dec 23 2012
%F For n >= 1, a(n) = U(n-1,13/2), where U(k,x) represents Chebyshev polynomial of the second order. - _Milan Janjic_, Jan 23 2015
%F a(n) = sqrt((A078367(n+1)^2 - 4)/285), n>=0, (Pell equation d=285, +4).
%F E.g.f.: exp(17*x/2)*(sqrt(285)*cosh(sqrt(285)*x/2) + 17*sinh(sqrt(285)*x/2))/sqrt(285). - _Stefano Spezia_, Aug 19 2023
%t CoefficientList[Series[1/(1 - 17 x + x^2), {x, 0, 20}], x] (* _Vincenzo Librandi_, Dec 24 2012 *)
%t LinearRecurrence[{17,-1},{1,17},20] (* _Harvey P. Dale_, Aug 02 2018 *)
%o (Sage) [lucas_number1(n,17,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%o (Magma) I:=[1, 17, 288]; [n le 3 select I[n] else 17*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Dec 24 2012
%o (PARI) my(x='x+O('x^20)); Vec(1/(1-17*x+x^2)) \\ _G. C. Greubel_, May 25 2019
%o (GAP) a:=[1,17,288];; for n in [4..20] do a[n]:=17*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, May 25 2019
%Y Cf. A077428, A078355 (Pell +4 equations). Cf. A078367.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 29 2002
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