|
%I #55 Feb 26 2024 01:31:33
%S 0,0,0,0,1,3,6,10,15,21,29,39,51,65,81,99,120,144,171,201,234,270,310,
%T 354,402,454,510,570,635,705,780,860,945,1035,1131,1233,1341,1455,
%U 1575,1701,1834,1974,2121,2275,2436,2604,2780,2964,3156,3356
%N Partial sums of floor(n^2/12).
%C Partial sums of A008724.
%C 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
%H Vincenzo Librandi, <a href="/A175724/b175724.txt">Table of n, a(n) for n = 0..10000</a>
%H Allan Bickle and Zhongyuan Che, <a href="https://arxiv.org/abs/1908.09202">Wiener indices of maximal k-degenerate graphs</a>, arXiv:1908.09202 [math.CO], 2019.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a> J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,0,0,1,-3,3,-1).
%F a(n) = round((2*n^3 + 3*n^2 - 18*n)/72).
%F a(n) = a(n-6) + (n-2)*(n-3)/2, n>5.
%F 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.
%F G.f.: x^4/((x+1)*(x^2+x+1)*(x^2-x+1)*(x-1)^4).
%F An explicit formula appears in the Bickle/Che paper.
%p A175724 := proc(n) add( floor(i^2/12) ,i=0..n) ; end proc:
%t Accumulate[Floor[Range[0, 49]^2/12]]
%o (Magma) [ &+[ Floor(j^2/12): j in [0..n] ]: n in [0..60] ];
%o (PARI) vector(61, n, round((2*(n-1)^3 +3*(n-1)^2 -18*(n-1))/72) ) \\ _G. C. Greubel_, Dec 05 2019
%o (Sage) [round((2*n^3 +3*n^2 -18*n)/72) for n in (0..60)] # _G. C. Greubel_, Dec 05 2019
%Y Cf. A008724.
%Y Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046, A122047, (this sequence).
%K nonn,easy
%O 0,6
%A _Mircea Merca_, Aug 18 2010
|
