1,2

If k is in the sequence, then it has successor 7*k + 4*sqrt{3*(k^2 - 1)}. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 28 2002

Chebyshev's polynomials T(n,x) evaluated at x=7.

a(n+1) give all (nontrivial) solutions of Pell equation a(n+1)^2 - 48*b(n+1)^2 = +1 with b(n+1)=A007655(n+2), n>=0.

Also numbers x of the form 3k+1 such that x^2 = 3n^2+1. Also solutions of x in x^2 - 3*y^2 = 1 in A001075 if x = 3k+1 k=1,2,... - Cino Hilliard (hillcino368(AT)gmail.com), Mar 05 2005

Equals sqrt(12*A011944(n)^2 + 1).

In addition to having integral standard deviation, these n consecutive integers also have integral mean. This question was posed by Jim Delany of Cal Poly in 1989. The solution appeared in the American Mathematical Monthly Vol. 97, No. 5, (May, 1990), pp. 432 as problem E3302. - Ronald S. Tiberio (chuck_tiberio(AT)wellesley.k12.ma.us), Jun 23 2008

Lebl and Lichtblau give the formula a(d) = ((7+4*sqrt(3))^d + (7-4*sqrt(3))^d)/2 in Theorem 1.2(iii), p.4. Abstract: Let p(x,y) be a polynomial of degree d with N positive coefficients and no negative coefficients, such that p=1 when x+y=1. It is known that the sharp estimate d <= 2N-3 holds. In this paper we study the p that minimize N and we give complete classification of these polynomials up to d=17 by computational methods. We use a linear algebra approach and a mixed linear programming approach. The question is motivated by a question in CR geometry. In particular, a complete classification of polynomials minimizing N is an important first step in the complete classification of CR maps of spheres in different dimensions. - Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 05 2008

E. K. Lloyd "The standard deviation of 1, 2, .., n, Pell's equation and rational triangles", preprint.

Tanya Khovanova, Recursive Sequences

Jiri Lebl and Daniel Lichtblau, Uniqueness of certain polynomials constant on a hyperplane

Index entries for sequences related to Chebyshev polynomials.

Index entries for sequences related to linear recurrences with constant coefficients

a(m) = 14a(m-1) - a(m-2).

a(n) ~ (1/2)*(2 + sqrt(3))^(2*n) - Joe Keane (jgk(AT)jgk.org), May 15 2002

a(n) = T(n, 7) = (S(n, 14)-S(n-2, 14))/2 = T(2*n, 2) with S(n, x) := U(n, x/2) and T(n, x), resp. U(n, x), are Chebyshev's polynomials of the first, resp. second, kind. See A053120 and A049310. S(-2, x) := -1, S(-1, x) := 0, S(n, 14)=A007655(n+2).

a(n) = ((7+4*sqrt(3))^n + (7-4*sqrt(3))^n)/2.

a(n) = sqrt(48*A007655(n+1)^2 + 1).

G.f.: (1-7*x)/(1-14*x+x^2).

a(n) = Cosh[2n*ArcSinh[Sqrt[3]]] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008

a(n) = ((7+4*sqrt(3))^n + (7-4*sqrt(3))^n)/2. - Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 05 2008

(PARI) a(n)=if(n<0, 0, subst(poltchebi(n), x, 7))

(PARI) g(n) = forstep(x=1, n, 3, y=(x^2-1)/3; if(issquare(y), print1(x", "))) (Hilliard)

a(n)=A001075(2n)

Cf. A007654, A011944.

Sequence in context: A093172 A074110 A155644 this_sequence A083083 A022007 A058805

Adjacent sequences: A011940 A011941 A011942 this_sequence A011944 A011945 A011946

nonn,easy

E. K. Lloyd

Better description from Lekraj Beedassy (blekraj(AT)yahoo.com), Jun 27 2002

Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Nov 08 2002