A007009 Number of 3-voter voting schemes with n linearly ranked choices. (Formerly M3435)

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

Offset

1,2

Comments

With a(0) = 0 nontrivial integer solutions of (x + y)^3 = (x - y)^4. If x = a(n) then y = a(n + (-1)^n). - Thomas Scheuerle, Mar 22 2023

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)*ceiling(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, 1, `if`(j=1, [4, -4, -4, 10, -4, -4, 4, -1][i], 0)))^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: A213760 A062479 A327032 * 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