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 A007654 Numbers n such that the standard deviation of 1,...,n is an integer. (Formerly M3154) 15
 0, 3, 48, 675, 9408, 131043, 1825200, 25421763, 354079488, 4931691075, 68689595568, 956722646883, 13325427460800, 185599261804323, 2585064237799728, 36005300067391875, 501489136705686528, 6984842613812219523, 97286307456665386800, 1355023461779503195683 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Gives solutions to the Diophantine equation k=sqrt[n*(n+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 REFERENCES 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). 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 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. A007655. Sequence in context: A294829 A264730 A024042 * A001080 A099852 A270005 Adjacent sequences:  A007651 A007652 A007653 * A007655 A007656 A007657 KEYWORD easy,nonn AUTHOR EXTENSIONS Corrected by Keith Lloyd, Mar 15 1996 STATUS approved

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Last modified October 16 13:30 EDT 2019. Contains 328092 sequences. (Running on oeis4.)