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A011943
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Numbers n such that any group of n consecutive integers has integral standard deviation {viz. A011944(n)}.
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15
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1, 7, 97, 1351, 18817, 262087, 3650401, 50843527, 708158977, 9863382151, 137379191137, 1913445293767, 26650854921601, 371198523608647, 5170128475599457, 72010600134783751, 1002978273411373057
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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. - Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 05 2008
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REFERENCES
| E. K. Lloyd "The standard deviation of 1, 2, .., n, Pell's equation and rational triangles", preprint.
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LINKS
| 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
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FORMULA
| 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
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PROG
| (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)
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CROSSREFS
| a(n)=A001075(2n)
Row 2 of array A188644
Cf. A007654, A011944.
Sequence in context: A093172 A074110 A155644 * A188441 A178808 A083083
Adjacent sequences: A011940 A011941 A011942 * A011944 A011945 A011946
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KEYWORD
| nonn,easy
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AUTHOR
| E. K. Lloyd
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EXTENSIONS
| 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
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