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A057081 Even-indexed Chebyshev U-polynomials evaluated at sqrt(11)/2.

%I A057081 #77 Jun 26 2025 07:32:47
%S A057081 1,10,89,791,7030,62479,555281,4935050,43860169,389806471,3464398070,
%T A057081 30789776159,273643587361,2432002510090,21614379003449,
%U A057081 192097408520951,1707262297685110,15173263270645039,134852107138120241,1198495700972437130,10651609201613813929
%N A057081 Even-indexed Chebyshev U-polynomials evaluated at sqrt(11)/2.
%C A057081 This is the m=11 member of the m-family of sequences S(n,m-2)+S(n-1,m-2) = S(2*n,sqrt(m)) (for S(n,x) see Formula). The m=4..10 instances are A005408, A002878, A001834, A030221, A002315, A033890 and A057080, resp. The m=1..3 (signed) sequences are: A057078, A057077 and A057079, resp.
%C A057081 General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim_{n->oo} a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. Examples in OEIS: a(1)=4 gives A002878. a(1)=5 gives A001834. a(1)=6 gives A030221. a(1)=7 gives A002315. a(1)=8 gives A033890. a(1)=9 gives A057080. a(1)=10 gives A057081. - _Ctibor O. Zizka_, Sep 02 2008
%C A057081 The primes in this sequence are 89, 389806471, 192097408520951, 7477414486269626733119, ... - _Ctibor O. Zizka_, Sep 02 2008
%C A057081 The aerated sequence (b(n))n>=1 = [1, 0, 10, 0, 89, 0, 791, 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 = -7, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - _Peter Bala_, Mar 22 2015
%H A057081 G. C. Greubel, <a href="/A057081/b057081.txt">Table of n, a(n) for n = 0..1000</a>
%H A057081 Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.
%H A057081 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 A057081 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 A057081 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A057081 Wolfdieter 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), rhs, m=11.
%H A057081 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 A057081 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 A057081 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A057081 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,-1).
%F A057081 a(n) = 9*a(n-1) - a(n-2), a(-1)=-1, a(0)=1.
%F A057081 a(n) = S(n, 9) + S(n-1, 9) = S(2*n, sqrt(11)) with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 9) = A018913(n).
%F A057081 G.f.: (1+x)/(1-9*x+x^2).
%F A057081 Let q(n, x) = Sum{i=0..n} x^(n-i)*binomial(2*n-i, i), a(n) = (-1)^n*q(n, -11). - _Benoit Cloitre_, Nov 10 2002
%F A057081 a(n) = L(n,-9)*(-1)^n, where L is defined as in A108299; see also A070998 for L(n,+9). - _Reinhard Zumkeller_, Jun 01 2005
%F A057081 From _Peter Bala_, Jun 08 2025: (Start)
%F A057081 a(n) = (1/sqrt(7)) * ( ((sqrt(11) + sqrt(7))/2)^(2*n+1) - ((sqrt(11) - sqrt(7))/2)^(2*n+1) ).
%F A057081 Sum_{n >= 1} (-1)^(n+1)/(a(n) - 1/a(n)) = 1/11 (telescoping series: 11/(a(n) - 1/a(n)) = 1/A018913(n+1) + 1/A018913(n)).
%F A057081 Conjecture: for k >= 1, Sum_{n >= 1} (-1)^(n+1)/(a(k*n) - s(k)/a(k*n)) = 1/(1 + a(k)) where s(k) = a(0) + a(1) + ... + a(k-1).
%F A057081 Product_{n >= 1} (a(n) + 1)/(a(n) - 1) = sqrt(11/7) [telescoping product: ((a(n) + 1)/(a(n) - 1))^2 = (1 - 4/b(n+1))/(1 - 4/b(n)), where b(n) = 2 + A056918(n)]. (End)
%p A057081 A057081 := proc(n)
%p A057081     option remember;
%p A057081     if n <= 1 then
%p A057081         op(n+1,[1,10]);
%p A057081     else
%p A057081         9*procname(n-1)-procname(n-2) ;
%p A057081     end if;
%p A057081 end proc: # _R. J. Mathar_, Apr 30 2017
%t A057081 CoefficientList[Series[(1 + x)/(1 - 9*x + x^2), {x,0,50}], x] (* or *) LinearRecurrence[{9,-1}, {1,10}, 50] (* _G. C. Greubel_, Apr 12 2017 *)
%o A057081 (Sage) [(lucas_number2(n,9,1)-lucas_number2(n-1,9,1))/7 for n in range(1, 20)] # _Zerinvary Lajos_, Nov 10 2009
%o A057081 (PARI) Vec((1+x)/(1-9*x+x^2) + O(x^30)) \\ _Michel Marcus_, Mar 22 2015
%Y A057081 Cf. A005408, A002878, A001834, A030221, A002315, A033890, A057080.
%Y A057081 Cf. A057078, A057077, A057079.
%Y A057081 Cf. A018913, A049310.
%Y A057081 Cf. A100047.
%K A057081 nonn,easy
%O A057081 0,2
%A A057081 _Wolfdieter Lang_, Aug 04 2000