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A057080 Even-indexed Chebyshev U-polynomials evaluated at sqrt(10)/2. 24
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, 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,...}. [From 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

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

Merlini, Donatella, and Renzo Sprugnoli. "Arithmetic into geometric progressions through Riordan arrays." Discrete Mathematics 340.2 (2017): 160-174.

LINKS

Table of n, a(n) for n=0..19.

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.

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

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 * A001706 A251284 A144745

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 March 26 16:31 EDT 2017. Contains 284137 sequences.