A175724 Partial sums of floor(n^2/12).

0, 0, 0, 0, 1, 3, 6, 10, 15, 21, 29, 39, 51, 65, 81, 99, 120, 144, 171, 201, 234, 270, 310, 354, 402, 454, 510, 570, 635, 705, 780, 860, 945, 1035, 1131, 1233, 1341, 1455, 1575, 1701, 1834, 1974, 2121, 2275, 2436, 2604, 2780, 2964, 3156, 3356

Offset: 0

Author

Mircea Merca, Aug 18 2010

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A008724.

Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046, A122047, (this sequence).

Programs

Magma

[ &+[ Floor(j^2/12): j in [0..n] ]: n in [0..60] ];

Maple

A175724 := proc(n) add( floor(i^2/12) ,i=0..n) ; end proc:

Mathematica

Accumulate[Floor[Range[0, 49]^2/12]]

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

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.