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%I #50 Dec 07 2019 12:18:24
%S 1,19,360,6821,129239,2448720,46396441,879083659,16656193080,
%T 315588584861,5979526919279,113295422881440,2146633507828081,
%U 40672741225852099,770635449783361800,14601400804658022101
%N A Chebyshev S-sequence with Diophantine property.
%C a(n) gives the general (positive integer) solution of the Pell equation b^2 - 357*a^2 =+4 with companion sequence b(n)=A078369(n+1), n>=0.
%C This is the m=21 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..20 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190, A004191, A078362, A007655, A078364, A077412, A078366 and A049660. 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 19'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,...,18}. _Milan Janjic_, Jan 25 2015
%H Vincenzo Librandi, <a href="/A078368/b078368.txt">Table of n, a(n) for n = 0..200</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=19, q=-1.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H W. 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=21.
%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 (19,-1).
%F a(n) = 19*a(n-1)-a(n-2), n >= 1; a(-1)=0, a(0)=1.
%F a(n) = (ap^(n+1)-am^(n+1))/(ap-am) with ap = (19+sqrt(357))/2 and am = (19-sqrt(357))/2.
%F a(n) = S(2*n+1, sqrt(21))/sqrt(21) = S(n, 19); S(n, x) := U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
%F G.f.: 1/(1-19*x+x^2).
%F a(n) = Sum_{k=0..n} A101950(n,k)*18^k. - _Philippe Deléham_, Feb 10 2012
%F Product {n >= 0} (1 + 1/a(n)) = 1/17*(17 + sqrt(357)). - _Peter Bala_, Dec 23 2012
%F Product {n >= 1} (1 - 1/a(n)) = 1/38*(17 + sqrt(357)). - _Peter Bala_, Dec 23 2012
%t Join[{a=1,b=19},Table[c=19*b-a;a=b;b=c,{n,40}]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 14 2011 *)
%t LinearRecurrence[{19,-1},{1,19},20] (* _Harvey P. Dale_, Feb 10 2019 *)
%o (Sage) [lucas_number1(n,19,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%Y a(n) = sqrt((A078369(n+1)^2 - 4)/357), n>=0, (Pell equation d=357, +4).
%Y Cf. A077428, A078355 (Pell +4 equations).
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Nov 29 2002
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