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 A057081 Even indexed Chebyshev U-polynomials evaluated at sqrt(11)/2. 18
 1, 10, 89, 791, 7030, 62479, 555281, 4935050, 43860169, 389806471, 3464398070, 30789776159, 273643587361, 2432002510090, 21614379003449, 192097408520951, 1707262297685110, 15173263270645039, 134852107138120241 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. 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 General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity 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] The primes in this sequence are 89, 389806471, 192097408520951, 7477414486269626733119, ... - Ctibor O. Zizka, Sep 02 2008 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 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38. Andersen, K., Carbone, L. and Penta, D., Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9. Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13. Tanya Khovanova, Recursive Sequences Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), rhs, m=11. H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277. H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume. Index entries for linear recurrences with constant coefficients, signature (9,-1). FORMULA a(n) = 9*a(n-1)-a(n-2), a(-1)=-1, a(0)=1. 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). G.f.: (1+x)/(1-9*x+x^2). 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 MAPLE A057081 := proc(n)     option remember;     if n <= 1 then         op(n+1, [1, 10]);     else         9*procname(n-1)-procname(n-2) ;     end if; end proc: # R. J. Mathar, Apr 30 2017 MATHEMATICA 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 *) PROG (Sage) [(lucas_number2(n, 9, 1)-lucas_number2(n-1, 9, 1))/7 for n in range(1, 20)] # Zerinvary Lajos, Nov 10 2009 (PARI) Vec((1+x)/(1-9*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 22 2015 CROSSREFS Cf. A005408, A002878, A001834, A030221, A002315, A033890, A057080. Cf. A057078, A057077, A057079. Cf. A018913, A049310. Cf. A100047. Sequence in context: A000826 A282555 A120923 * A024132 A192898 A044261 Adjacent sequences:  A057078 A057079 A057080 * A057082 A057083 A057084 KEYWORD nonn,easy AUTHOR Wolfdieter Lang Aug 04 2000 STATUS approved

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Last modified September 21 12:49 EDT 2020. Contains 337272 sequences. (Running on oeis4.)