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A057081 Even indexed Chebyshev U-polynomials evaluated at sqrt(11)/2. 19
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, primes in it A121534. a(1)=5 gives A001834, primes in it A086386. a(1)=6 gives A030221, primes in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not in OEIS {389806471,192097408520951,...}. [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

REFERENCES

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

LINKS

Table of n, a(n) for n=0..18.

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=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 sequences related to Chebyshev polynomials.

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

PROG

(Sage) [(lucas_number2(n, 9, 1)-lucas_number2(n-1, 9, 1))/7 for n in xrange(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

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 March 29 15:31 EDT 2017. Contains 284273 sequences.