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) = (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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James Sellers, Sep 08 2000
STATUS
approved
