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 A057080 Even-indexed Chebyshev U-polynomials evaluated at sqrt(10)/2. 23
 1, 9, 71, 559, 4401, 34649, 272791, 2147679, 16908641, 133121449, 1048062951, 8251382159, 64962994321, 511452572409, 4026657584951, 31701808107199, 249587807272641, 1965000650073929, 15470417393318791, 121798338496476399 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) = L(n,-8)*(-1)^n, where L is defined as in A108299; see also A070997 for L(n,+8). - 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. [From Ctibor O. Zizka, Sep 02 2008] The primes in this sequence are 71, 34649, 16908641, 8251382159, 31701808107199,... - Ctibor O. Zizka, Sep 02 2008 The aerated sequence (b(n))n>=1 = [1, 0, 9, 0, 71, 0, 559, 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 = -6, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047. - Peter Bala, Mar 22 2015 REFERENCES Fink, Alex, Richard Guy, and Mark Krusemeyer. "Partitions with parts occurring at most thrice." Contributions to Discrete Mathematics 3.2 (2008), 76-114. See Section 13. 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. Tanya Khovanova, Recursive Sequences W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eq.(44), rhs, m=10. Donatella Merlini and Renzo Sprugnoli, Arithmetic into geometric progressions through Riordan arrays, Discrete Mathematics 340.2 (2017): 160-174. 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 (8,-1). FORMULA For all elements x of the sequence, 15*x^2 + 10 is a square. Lim. n-> Inf. a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson, Oct 13 2002 a(n) = 8*a(n-1)-a(n-2), a(-1)=-1, a(0)=1. a(n) = S(n, 8)+S(n-1, 8) = S(2*n, sqrt(10)) with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 8) = A001090(n). G.f.: (1+x)/(1-8*x+x^2). a(n) = [ [(4+sqrt(15))^(n+1) - (4-sqrt(15))^(n+1)] + [(4+sqrt(15))^n - (4-sqrt(15))^n] ] / (2*sqrt(15)). - Gregory V. Richardson, Oct 13 2002 a(n) = sqrt((5*A070997(n)^2 - 2)/3) (cf. Richardson comment). Let q(n, x) = sum(i=0, n, x^(n-i)*binomial(2*n-i, i)) then a(n) = (-1)^n*q(n,-10). - Benoit Cloitre, Nov 10 2002 a(n) = Jacobi_P(n,1/2,-1/2,4)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry, Feb 03 2006 a(n+1) = 4*a(n)+((3*a(n)^2+2)*5)^0.5. - Richard Choulet, Aug 30 2007 MAPLE A057080 := proc(n)     option remember;     if n <= 1 then         op(n+1, [1, 9]);     else         8*procname(n-1)-procname(n-2) ;     end if; end proc: # R. J. Mathar, Apr 30 2017 MATHEMATICA CoefficientList[Series[(1 + x) / (1 - 8 x + x^2), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 22 2015 *) PROG (Sage) [(lucas_number2(n, 8, 1)-lucas_number2(n-1, 8, 1))/6 for n in xrange(1, 21)] # Zerinvary Lajos, Nov 10 2009 (PARI) Vec((1+x)/(1-8*x+x^2) + O(x^30)) \\ Michel Marcus, Mar 22 2015 (MAGMA) I:=[1, 9]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 22 2015 CROSSREFS Cf. A033890, A100047. Sequence in context: A081900 A164551 A178869 * A287819 A001706 A251284 Adjacent sequences:  A057077 A057078 A057079 * A057081 A057082 A057083 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 04 2000 STATUS approved

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Last modified October 23 14:43 EDT 2019. Contains 328345 sequences. (Running on oeis4.)