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)gmail.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

Index entries for sequences related to linear recurrences with constant coefficients

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

A054320(n) = A142238(2n) = A041006(2n)/2 = A041038(2n)/4 [From M. F. Hasler (MHasler(AT)univ-ag.fr), Feb 14 2009]

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

q=12; s=0; lst={}; Do[s+=n; If[Sqrt[q*s+1]==Floor[Sqrt[q*s+1]], AppendTo[lst, Sqrt[q*s+1]]], {n, 0, 9!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 02 2009]

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

(Other) sage: [(lucas_number2(n, 10, 1)-lucas_number2(n-1, 10, 1))/8 for n in xrange(1, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 10 2009]

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: A125423 A165149 A048346 this_sequence A124290 A094703 A144744

Adjacent sequences: A054317 A054318 A054319 this_sequence A054321 A054322 A054323

easy,nonn,new

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