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A007009 Number of 3-voter voting schemes with n linearly ranked choices.
(Formerly M3435)
4
1, 4, 12, 27, 54, 96, 160, 250, 375, 540, 756, 1029, 1372, 1792, 2304, 2916, 3645, 4500, 5500, 6655, 7986, 9504, 11232, 13182, 15379, 17836, 20580, 23625, 27000, 30720, 34816, 39304, 44217, 49572, 55404, 61731, 68590, 76000, 84000, 92610, 101871, 111804 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [broken link]

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

FORMULA

G.f.: x*(1-x^3)/((1-x)^4*(1-x^2)^2) = x*(1+x+x^2)/((1-x)^5*(1+x)^2).

a(n) = (1/2)*Sum_{k=1..n+1} k*floor(k/2)*ceil(k/2). - Vladeta Jovovic, Apr 29 2006

a(n) = A006009(n)/4.

a(n) = A007590(n+2)*A007590(n+1)/8. - Richard R. Forberg, Dec 03 2013

For n>1, a(n) = A000332(n+3) - A002624(n-2). - Antal Pinter, Sep 20 2015

a(n) = (n^4+6*n^3+12*n^2+8*n)/32 for n even; a(n) = (n^4+6*n^3+12*n^2+10*n+3)/32 for n odd. - Colin Barker, Jan 07 2016

MAPLE

a := n-> (Matrix([[0$4, 1, 4, 12, 27]]). Matrix(8, (i, j)-> if (i=j-1) then 1 elif j=1 then [4, -4, -4, 10, -4, -4, 4, -1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=1..40); # Alois P. Heinz, Aug 13 2008

MATHEMATICA

LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {1, 4, 12, 27, 54, 96, 160}, 50] (* Vincenzo Librandi, Sep 21 2015 *)

PROG

(MAGMA) I:=[1, 4, 12, 27, 54, 96, 160]; [n le 7 select I[n] else 3*Self(n-1)-Self(n-2)- 5*Self(n-3)+5*Self(n-4)+Self(n-5)-3*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Sep 21 2015

(PARI) Vec(x*(1-x^3)/((1-x)^4*(1-x^2)^2) + O(x^100)) \\ Colin Barker, Jan 07 2016

CROSSREFS

Cf. A034828 (first differences).

Sequence in context: A104385 A213760 A062479 * A188814 A104384 A013697

Adjacent sequences:  A007006 A007007 A007008 * A007010 A007011 A007012

KEYWORD

nonn,easy

AUTHOR

Daniel E. Loeb

EXTENSIONS

More terms from James A. Sellers, Sep 08 2000

STATUS

approved

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Last modified April 20 06:36 EDT 2019. Contains 322294 sequences. (Running on oeis4.)