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A007009 Number of 3-voter voting schemes with n linearly ranked choices.

%I A007009 M3435 #58 Jul 02 2025 16:01:55
%S A007009 1,4,12,27,54,96,160,250,375,540,756,1029,1372,1792,2304,2916,3645,
%T A007009 4500,5500,6655,7986,9504,11232,13182,15379,17836,20580,23625,27000,
%U A007009 30720,34816,39304,44217,49572,55404,61731,68590,76000,84000,92610,101871,111804
%N A007009 Number of 3-voter voting schemes with n linearly ranked choices.
%C A007009 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
%D A007009 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007009 Colin Barker, <a href="/A007009/b007009.txt">Table of n, a(n) for n = 1..1000</a>
%H A007009 Daniel E. Loeb, <a href="http://www.labri.u-bordeaux.fr/~loeb/vote.html">On Games, Voting Schemes and Distributive Lattices</a>. LaBRI Report 625-93, University of Bordeaux I, 1993. [broken link]
%H A007009 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A007009 G.f.: x*(1-x^3)/((1-x)^4*(1-x^2)^2) = x*(1+x+x^2)/((1-x)^5*(1+x)^2).
%F A007009 a(n) = (1/2)*Sum_{k=1..n+1} k*floor(k/2)*ceiling(k/2). - _Vladeta Jovovic_, Apr 29 2006
%F A007009 a(n) = A006009(n)/4.
%F A007009 a(n) = A007590(n+2)*A007590(n+1)/8. - _Richard R. Forberg_, Dec 03 2013
%F A007009 For n > 1, a(n) = A000332(n+3) - A002624(n-2). - _Antal Pinter_, Sep 20 2015
%F A007009 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
%p A007009 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]:
%p A007009 seq(a(n), n=1..40);  # _Alois P. Heinz_, Aug 13 2008
%t A007009 LinearRecurrence[{3, -1, -5, 5, 1, -3, 1}, {1, 4, 12, 27, 54, 96, 160}, 50] (* _Vincenzo Librandi_, Sep 21 2015 *)
%o A007009 (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
%o A007009 (PARI) Vec(x*(1-x^3)/((1-x)^4*(1-x^2)^2) + O(x^100)) \\ _Colin Barker_, Jan 07 2016
%Y A007009 Cf. A034828 (first differences).
%K A007009 nonn,easy
%O A007009 1,2
%A A007009 _Daniel E. Loeb_
%E A007009 More terms from _James Sellers_, Sep 08 2000