OFFSET
0,6
COMMENTS
Partial sums of A008724.
Maximum Wiener index of all maximal 6-degenerate graphs with n-2 vertices. (A maximal 6-degenerate graph can be constructed from a 6-clique by iteratively adding a new 6-leaf (vertex of degree 6) adjacent to 6 existing vertices.) The extremal graphs are 6th powers of paths, so the bound also applies to 6-trees. - Allan Bickle, Sep 18 2022
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,1,-3,3,-1).
FORMULA
a(n) = round((2*n^3 + 3*n^2 - 18*n)/72).
a(n) = a(n-6) + (n-2)*(n-3)/2, n>5.
a(n) = 3*a(n-1) -3*a(n-2) +a(n-3) +a(n-6) -3*a(n-7) +3*a(n-8) -a(n-9), n>8.
G.f.: x^4/((x+1)*(x^2+x+1)*(x^2-x+1)*(x-1)^4).
An explicit formula appears in the Bickle/Che paper.
MAPLE
A175724 := proc(n) add( floor(i^2/12) , i=0..n) ; end proc:
MATHEMATICA
Accumulate[Floor[Range[0, 49]^2/12]]
PROG
(Magma) [ &+[ Floor(j^2/12): j in [0..n] ]: n in [0..60] ];
(PARI) vector(61, n, round((2*(n-1)^3 +3*(n-1)^2 -18*(n-1))/72) ) \\ G. C. Greubel, Dec 05 2019
(Sage) [round((2*n^3 +3*n^2 -18*n)/72) for n in (0..60)] # G. C. Greubel, Dec 05 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mircea Merca, Aug 18 2010
STATUS
approved