0,2

a(n) = L(n,-8)*(-1)^n, where L is defined as in A108299; see also A070997 for L(n,+8). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (ctibor.zizka(AT)seznam.cz), Sep 02 2008]

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), rhs, m=10.

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

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 (omomom(AT)hotmail.com), 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 (omomom(AT)hotmail.com), Oct 13 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -10)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002

a(n)=Jacobi_P(n,1/2,-1/2,4)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), Feb 03 2006

a(n+1)=4*a(n)+((3*a(n)^2+2)*5)^0.5. - Richard Choulet (richardchoulet(AT)yahoo.fr), Aug 30 2007

(Other) sage: [(lucas_number2(n, 8, 1)-lucas_number2(n-1, 8, 1))/6 for n in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 2009]

A033890. a(n)=sqrt((5*A070997(n)^2 - 2)/3) (cf. Richardson comment).

Sequence in context: A156705 A081900 A164551 this_sequence A001706 A158193 A123987

Adjacent sequences: A057077 A057078 A057079 this_sequence A057081 A057082 A057083

nonn,new

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 04 2000