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A057080 Even-indexed Chebyshev U-polynomials evaluated at sqrt(10)/2. 22
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

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.

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 sequences related to Chebyshev polynomials.

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 August 17 07:46 EDT 2017. Contains 290635 sequences.