0,2

A Pellian sequence.

In general, sum{k=0..n, binomial(2n-k,k)j^(n-k)}=(-1)^n*U(2n,I*sqrt(j)/2), I=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005

a(n) = L(n,8), where L is defined as in A108299; see also A057080 for L(n,-8). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

Number of 01-avoiding words of length n on alphabet {0,1,2,3,4,5,6,7} which do not end in 0. - Tanya Khovanova (tanyakh(AT)yahoo.com), Jan 10 2007

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

For all members x of the sequence, 15*x^2 - 6 is a square. Lim. n-> Inf. a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 12 2002

a(n) = (5+sqrt(15))/10 * (4+sqrt(15))^n + (5-sqrt(15))/10 * (4-sqrt(15))^n

a(n) ~ 1/10*sqrt(10)*(1/2*(sqrt(10)+sqrt(6)))^(2*n+1)

a(n) = U(n, 4)-U(n-1, 4) = T(2*n+1, sqrt(5/2))/sqrt(5/2), with Chebyshev's U and T Polynomials and U(-1, x) := 0. U(n, 4)=A001090(n+1), n>=-1.

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 6)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002

a(n)a(n+3) = 48 + a(n+1)a(n+2). - R. Stephan, May 29 2004

a(n)=(-1)^n*U(2n, I*sqrt(6)/2), U(n, x) Chebyshev polynomial of second kind, I=sqrt(-1); - Paul Barry (pbarry(AT)wit.ie), Mar 13 2005

G.f.: (1-x)/(1-8*x+x^2). a(n)=a(-1-n).

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), A001090(n+1)] = [1,6; 1,7]^(n+1) * [1,0]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 21 2008

(PARI) a(n)=subst(9*poltchebi(n)-poltchebi(n-1), x, 4)/5 /* Michael Somos Jun 07 2005 */

(PARI) a(n)=if(n<0, n=-1-n); polcoeff((1-x)/(1-8*x+x^2)+x*O(x^n), n) /* Michael Somos Jun 07 2005 */

a(n) = sqrt((3*A057080(n)^2+2)/5) (cf. Richardson comment)

Cf. A057080, A001090, A001091.

Row 8 of array A094954.

Cf. A001090.

Adjacent sequences: A070994 A070995 A070996 this_sequence A070998 A070999 A071000

Sequence in context: A121183 A069404 A015564 this_sequence A122372 A083068 A097189

nonn

Joe Keane (jgk(AT)jgk.org), May 18 2002