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A078362 A Chebyshev S-sequence with Diophantine property.

%I A078362 #74 Jan 05 2025 19:51:37
%S A078362 1,13,168,2171,28055,362544,4685017,60542677,782369784,10110264515,
%T A078362 130651068911,1688353631328,21817946138353,281944946167261,
%U A078362 3643466354036040,47083117656301259,608437063177880327
%N A078362 A Chebyshev S-sequence with Diophantine property.
%C A078362 a(n) gives the general (positive integer) solution of the Pell equation b^2 - 165*a^2 = +4 with companion sequence b(n)=A078363(n+1), n >= 0.
%C A078362 This is the m=15 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..14 (nonnegative) sequences are: A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913, A004189, A004190 and A004191. The m=1..3 (signed) sequences are A049347, A056594, A010892.
%C A078362 For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 13'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 A078362 For n >= 1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,12}. - _Milan Janjic_, Jan 23 2015
%H A078362 Vincenzo Librandi, <a href="/A078362/b078362.txt">Table of n, a(n) for n = 0..900</a>
%H A078362 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://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=13, q=-1.
%H A078362 M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
%H A078362 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A078362 W. Lang, <a href="https://web.archive.org/web/2024*/https://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=15.
%H A078362 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A078362 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-1).
%F A078362 a(n) = 13*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1.
%F A078362 a(n) = S(2*n+1, sqrt(15))/sqrt(15) = S(n, 13), where S(n, x) = U(n, x/2), Chebyshev polynomials of the 2nd kind, A049310.
%F A078362 a(n) = (ap^(n+1) - am^(n+1))/(ap-am) with ap = (13+sqrt(165))/2 and am = (13-sqrt(165))/2.
%F A078362 G.f.: 1/(1 - 13*x + x^2).
%F A078362 a(n) = Sum_{k=0..n} A101950(n,k)*12^k. - _Philippe Deléham_, Feb 10 2012
%F A078362 Product {n >= 0} (1 + 1/a(n)) = (1/11)*(11 + sqrt(165)). - _Peter Bala_, Dec 23 2012
%F A078362 Product {n >= 1} (1 - 1/a(n)) = (1/26)*(11 + sqrt(165)). - _Peter Bala_, Dec 23 2012
%F A078362 For n >= 1, a(n) = U(n-1,13/2), where U(k,x) represents Chebyshev polynomial of the second order.
%F A078362 a(n) = sqrt((A078363(n+1)^2 - 4)/165), n>=0, (Pell equation d=165, +4).
%t A078362 CoefficientList[Series[1/(1 - 13 x + x^2), {x, 0, 20}], x] (* _Vincenzo Librandi_, Dec 24 2012 *)
%t A078362 LinearRecurrence[{13,-1},{1,13},20] (* _Harvey P. Dale_, Feb 07 2019 *)
%o A078362 (Sage) [lucas_number1(n,13,1) for n in range(1,20)] # _Zerinvary Lajos_, Jun 25 2008
%o A078362 (Magma) I:=[1, 13, 168]; [n le 3 select I[n] else 13*Self(n-1)-Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Dec 24 2012
%o A078362 (PARI) my(x='x+O('x^20)); Vec(1/(1-13*x+x^2)) \\ _G. C. Greubel_, May 25 2019
%o A078362 (GAP) a:=[1,13,168];; for n in [4..20] do a[n]:=13*a[n-1]-a[n-2]; od; a; # _G. C. Greubel_, May 25 2019
%Y A078362 Cf. A078363.
%K A078362 nonn,easy
%O A078362 0,2
%A A078362 _Wolfdieter Lang_, Nov 29 2002