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A007654
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Numbers n such that the standard deviation of 1,...,n is an integer.
(Formerly M3154)
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15
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0, 3, 48, 675, 9408, 131043, 1825200, 25421763, 354079488, 4931691075, 68689595568, 956722646883, 13325427460800, 185599261804323, 2585064237799728, 36005300067391875, 501489136705686528, 6984842613812219523, 97286307456665386800, 1355023461779503195683
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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D. A. Benaron, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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).
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A007655.
Sequence in context: A294829 A264730 A024042 * A001080 A099852 A270005
Adjacent sequences: A007651 A007652 A007653 * A007655 A007656 A007657
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KEYWORD
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easy,nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Corrected by Keith Lloyd, Mar 15 1996
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STATUS
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approved
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