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A077416 Chebyshev S-sequence with Diophantine property.

%I A077416 #48 May 13 2025 00:56:26
%S A077416 1,13,155,1847,22009,262261,3125123,37239215,443745457,5287706269,
%T A077416 63008729771,750817050983,8946795882025,106610733533317,
%U A077416 1270382006517779,15137973344680031,180385298129642593
%N A077416 Chebyshev S-sequence with Diophantine property.
%C A077416 7*b(n)^2 - 5*a(n)^2 = 2 with companion sequence b(n) = A077417(n), n>=0.
%C A077416 a(n) = L(n,-12)*(-1)^n, where L is defined as in A108299; see also A077417 for L(n,+12). - _Reinhard Zumkeller_, Jun 01 2005
%C A077416 The aerated sequence (b(n))n>=1 = [1, 0, 13, 0, 155, 0, 1857, 0, ...] is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -10, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - _Peter Bala_, May 12 2025
%H A077416 Ivan Panchenko, <a href="/A077416/b077416.txt">Table of n, a(n) for n = 0..200</a>
%H A077416 K. Andersen, L. Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
%H A077416 Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
%H A077416 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A077416 H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
%H A077416 H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
%H A077416 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A077416 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (12,-1).
%F A077416 a(n) = 12*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.
%F A077416 a(n) = S(n, 12) + S(n-1, 12) = S(2*n, sqrt(14)) with S(n, x) := U(n, x/2) Chebyshev's polynomials of the second kind. See A049310. S(-1, x)=0, S(n, 12) = A004191(n).
%F A077416 G.f.: (1+x)/(1-12*x+x^2).
%F A077416 a(n) = (ap^(2*n+1) - am^(2*n+1))/(ap - am) with ap := (sqrt(7)+sqrt(5))/sqrt(2) and am := (sqrt(7)-sqrt(5))/sqrt(2).
%F A077416 a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n-k,k) * 14^(n-k).
%F A077416 a(n) = sqrt((7*A077417(n)^2 - 2)/5).
%F A077416 From _Peter Bala_, May 09 2025: (Start)
%F A077416 a(n) = Dir(n, 6), where Dir(n, x) denotes the n-th row polynomial of the triangle A244419.
%F A077416 a(n)^2 - 12*a(n)*a(n+1) + a(n+1)^2 = 14.
%F A077416 More generally, for real x, a(n+x)^2 - 12*a(n+x)*a(n+x+1) + a(n+x+1)^2 = 14, where a(n) := (ap^(2*n+1) - am^(2*n+1))/(ap - am), ap := sqrt(7/2) + sqrt(5/2) and am := sqrt(7/2) - sqrt(5/2), as given above.
%F A077416 Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/14 (telescoping series).
%F A077416 Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(7/5) (telescoping product). (End)
%t A077416 LinearRecurrence[{12,-1},{1,13},30] (* _Harvey P. Dale_, Apr 03 2013 *)
%o A077416 (Sage) [(lucas_number2(n,12,1)-lucas_number2(n-1,12,1))/10 for n in range(1, 18)] # _Zerinvary Lajos_, Nov 10 2009
%o A077416 (PARI) x='x+O('x^30); Vec((1+x)/(1-12*x+x^2)) \\ _G. C. Greubel_, Jan 18 2018
%o A077416 (Magma) I:=[1, 13]; [n le 2 select I[n] else 12*Self(n-1) - Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 18 2018
%Y A077416 Cf. A054320(n-1) with companion A072256(n), n>=1.
%K A077416 nonn,easy
%O A077416 0,2
%A A077416 _Wolfdieter Lang_, Nov 29 2002