0,2

Chebyshev's even indexed U-polynomials evaluated at sqrt(3).

a(n)^2 is a star number (A003154).

a(n) = L(n,-10)*(-1)^n, where L is defined as in A108299; see also A072256 for L(n,+10). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

(sqrt(2)+sqrt(3))^(2*n+1)=a(n)*sqrt(2)+A138288(n)*sqrt(3); a(n)=A138288(n)+A001078(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

Eric Weisstein's World of Mathematics, Star Number

(a(n)-1)^2+a(n)^2+(a(n)+1)^2=b(n)^2+(b(n)+1)^2=c(n), where b(n) is A031138 and c(n) is A007667

Any k in the sequence has the successor 5*k + 2sqrt{3(2*k^2 + 1)}. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 08 2002

a(n) = 10*a(n-1) - a(n-2); a(n)=(sqrt(6) - 2)/4*(5 + 2*sqrt(6))^n - (sqrt(6) + 2)/4*(5 - 2*sqrt(6))^n.

a(n) = U(2*(n-1), sqrt(3)) = S(n-1, 10) + S(n-2, 10) with Chebyshev's U(n, x) and S(n, x) := U(n, x/2) polynomials, and S(-1, x) := 0. S(n, 10) = A004189(n+1), n>=0.

For all members x of the sequence, 6*x^2 + 3 is a square. Lim. n-> Inf. a(n)/a(n-1) = 5 + 2*sqrt(6) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002

a(n) = [ [(5+2*sqrt(6))^n - (5-2*sqrt(6))^n] + [(5+2*sqrt(6))^(n-1) - (5-2*sqrt(6))^(n-1)] / (4*sqrt(6)) - 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, -12)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002

a(n) = A001079(n) + 3*A001078(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Mar 12 2008

a(1)^2=121 is the 5th star number (A003154).

(PARI) a(n)=if(n<1, 0, subst(poltchebi(n)-poltchebi(n-1), x, 5)/4)

A member of the family A057078, A057077, A057079, A005408, A002878, A001834, A030221, A002315, A033890, A057080, A057081, A054320, which are the expansions of (1+x) / (1-kx+x^2) with k = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 . Philippe DELEHAM, May 04 2004

Cf. A003154, A031138, A007667, A004189. a(n) = sqrt((3* A072256(n)^2 - 1)/2).

Cf. A138281.

Sequence in context: A142423 A125423 A048346 this_sequence A124290 A094703 A103542

Adjacent sequences: A054317 A054318 A054319 this_sequence A054321 A054322 A054323

easy,nonn

Ignacio Larrosa Canestro (ignacio.larrosa(AT)eresmas.net) Feb 27 2000

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Oct 31 2002