A049031 Maximization of sums of cubes of integer differences (b_[ i ]-i)^3 over permutations {b_[ i ], for i-1,2,...,n} on first n integers.

0, 0, 2, 8, 20, 40, 80, 140, 224, 336, 504, 720, 990, 1320, 1760, 2288, 2912, 3640, 4550, 5600, 6800, 8160, 9792, 11628, 13680, 15960, 18620, 21560, 24794, 28336, 32384, 36800, 41600, 46800, 52650, 58968, 65772, 73080, 81200, 89900

Offset

1,3

Links

Table of n, a(n) for n=1..40.

D. C. Blest, Optimising Sums of Cubes of Integer Differences, Math. Gaz., Vol. 84, No. 501 (Nov., 2000), pp. 509-513.

Formula

a[ n ] = (Sum_{i=1}^{i=r} [ (n+1-2i)^3-(n-r)*r^3 ]) / 3 where r=floor((n+2)/4)).

a(n) = r*(n-r+1)*(n-r)*(n-r-1)/3, where r = floor((n+1)/4). - Vladeta Jovovic, Dec 22 2004

G.f.: 2*x^3*(1+2*x+3*x^2+4*x^3+7*x^4+4*x^5+3*x^6+2*x^7+x^8)/((1+x)^3*(1+x^2)^3*(1-x)^5). - Vladeta Jovovic, Dec 22 2004

Mathematica

a[n_] := With[{r = Floor[(n+1)/4]}, r*(n-r+1)*(n-r)*(n-r-1)/3]; Array[a, 40] (* Jean-François Alcover, Sep 30 2016, after Vladeta Jovovic *)

Crossrefs

Cf. A049032.

Sequence in context: A294437 A128445 A007290 * A364583 A058037 A203420

Adjacent sequences: A049028 A049029 A049030 * A049032 A049033 A049034

Keyword

nonn,nice,easy

Author

David C Blest (D.Blest(AT)utas.edu.au)

Status

approved