A007654 Numbers k such that the standard deviation of 1,...,k is an integer. (Formerly M3154)

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

Offset

1,2

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).

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A294829 A264730 A024042 * A001080 A099852 A270005

Adjacent sequences: A007651 A007652 A007653 * A007655 A007656 A007657

Keyword

easy,nonn

Author

N. J. A. Sloane

Extensions

Corrected by Keith Lloyd, Mar 15 1996

Status

approved