A007654 Numbers k such that the standard deviation of 1,...,k is an integer.

0, 3, 48, 675, 9408, 131043, 1825200, 25421763, 354079488, 4931691075, 68689595568, 956722646883, 13325427460800, 185599261804323, 2585064237799728, 36005300067391875, 501489136705686528, 6984842613812219523, 97286307456665386800, 1355023461779503195683

Offset: 1

Author

N. J. A. Sloane

Keywords

• easy
• nonn

Comments

Gives solutions k to the Diophantine equation m^2 = k*(k+1)/3. - Anton Lorenz Vrba (anton(AT)a-l-v.net), Jun 28 2005

If x=a(n), y=a(n+1), z=a(n+2) are three consecutive terms, then x^2 - 16*y*x + 14*x*z + 16*y^2 - 16*z*y + z^2 = 144. The formula is symmetric in x and z, so it is also valid for x=a(n+2), y=a(n+1), z=a(n). - Alexander Samokrutov, Jul 02 2015

From Bernard Schott, Apr 09 2021 (Start):

Corresponding solutions m (of first comment) are in A011944.

Equivalently, numbers k such that k/3 and k+1 are both perfect squares. (End)

References

• Guy Alarcon and Yves Duval, TS: Préparation au Concours Général, RMS, Collection Excellence, Paris, 2010, chapitre 13, Questions proposées aux élèves de Terminale S, Exercice 1, p. 220, p. 223.
• D. A. Benaron, personal communication.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• Tanya Khovanova, Recursive Sequences
• E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.
• Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Crossrefs

Cf. A001075, A001353, A007655, A011944, A055793, A098301.

Programs

Magma

I:=[0,3]; [n le 2 select I[n] else 14*Self(n-1)-Self(n-2)+6: n in [1..20]]; // Vincenzo Librandi, Mar 05 2016

Mathematica

RecurrenceTable[{a[m] == 14 a[m - 1] - a[m - 2] + 6, a[1] == 0, a[2] == 3}, a, {m, 1, 17}] (* Michael De Vlieger, Jul 02 2015 *)
CoefficientList[Series[-3 x^2*(1 + x)/(-1 + x)/(1 - 14 x + x^2), {x, 0, 17}], x] (* Michael De Vlieger, Feb 02 2016 *)

PARI

concat(0,3*Vec((1+x)/(1-x)/(1-14*x+x^2)+O(x^98))) \\ Charles R Greathouse IV, May 14 2013

Formula

a(n) = 3*A098301(n-1).

a(m) = 14*a(m-1) - a(m-2) + 6.

G.f.: -3*x^2*(1+x)/(-1+x)/(1-14*x+x^2) = -3 + (1/2)/(-1+x) + (1/2)*(-97*x+7)/(1-14*x+x^2). - R. J. Mathar, Nov 20 2007

a(n) = (-2 + (7-4*sqrt(3))^n*(7+4*sqrt(3)) + (7-4*sqrt(3))*(7+4*sqrt(3))^n)/4. - Colin Barker, Mar 05 2016

From Bernard Schott, Apr 09 2021: (Start)

a(n) = 3 * A001353(n-1)^2.

a(n) = A055793(n+1) - 1 = A001075(n-1)^2 - 1. (End)

2*a(n) = A011943(n)-1. - R. J. Mathar, Mar 16 2023

Extensions

Corrected by Keith Lloyd, Mar 15 1996