%I #95 Sep 15 2024 06:51:12
%S 0,0,0,1,3,6,10,16,24,34,46,61,79,100,124,152,184,220,260,305,355,410,
%T 470,536,608,686,770,861,959,1064,1176,1296,1424,1560,1704,1857,2019,
%U 2190,2370,2560,2760,2970,3190,3421,3663,3916,4180,4456,4744,5044,5356,5681,6019,6370
%N Partial sums of floor(n^2/8).
%C Degree of the polynomial P(n+1,x), defined by P(n,x) = [x^(n-1)*P(n-1,x)*P(n-4,x)+P(n-2,x)*P(n-3,x)]/P(n-5,x) with P(1,x)=P(0,x)=P(-1,x)=P(-2,x)=P(-3,x)=1.
%C Define the sequence b(n) = 1, 4, 10, 20, 36, 60,... for n>=0 with g.f. 1/((1+x)*(1+x^2)*(1-x)^5). Then a(n+3) = b(n)-b(n-1) and b(n)+b(n+1)+b(n+2)+b(n+3) = A052762(n+7)/24. - _J. M. Bergot_, Aug 21 2013
%C Maximum Wiener index of all maximal 4-degenerate graphs with n-1 vertices. (A maximal 4-degenerate graph can be constructed from a 4-clique by iteratively adding a new 4-leaf (vertex of degree 4) adjacent to four existing vertices.) The extremal graphs are 4th powers of paths, so the bound also applies to 4-trees. - _Allan Bickle_, Sep 15 2022
%H Vincenzo Librandi, <a href="/A122046/b122046.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 A. N. W. Hone, <a href="/A122046/a122046.txt">Comments on A122046</a>
%H A. N. W. Hone, <a href="http://arXiv.org/abs/0807.2538">Algebraic curves, integer sequences and a discrete Painlevé transcendent</a>, Proceedings of SIDE 6, Helsinki, Finland, 2004; arXiv:0807.2538 [nlin.SI], 2008. [Set a(n)=d(n+3) on p. 8]
%H Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=4]
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1,1,-3,3,-1).
%F a(n) = Sum_{k=0..n} floor(k^2/8).
%F a(n) = round((2*n^3 + 3*n^2 - 8*n)/48) = round((4*n^3 + 6*n^2 - 16*n - 9)/96) = floor((2*n^3 + 3*n^2 - 8*n + 3)/48) = ceiling((2*n^3 + 3*n^2 - 8*n - 12)/48). - _Mircea Merca_
%F a(n) = a(n-8) + (n-4)^2 + n, n > 8. - _Mircea Merca_
%F From _Andrew Hone_, Jul 15 2008: (Start)
%F a(n+1) = cos((2*n+1)*Pi/4)/(4*sqrt(2)) + (2*n+3)*(2*n^2 + 6*n - 5)/96 + (-1)^n/32.
%F a(n+1) = A057077(n+1)/8 + A090294(n-1)/32 + (-1)^n/32.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7). (End)
%F O.g.f.: x^3 / ( (1+x)*(x^2+1)*(x-1)^4 ). - _R. J. Mathar_, Jul 15 2008
%F From _Johannes W. Meijer_, May 20 2011: (Start)
%F a(n+3) = A144678(n) + A144678(n-1) + A144678(n-2) + A144678(n-3);
%F a(n+3) = Sum_{k=0..6} min(6-k+1,k+1)* A190718(n+k-6). (End)
%F a(n) = (4*n^3 + 6*n^2 - 16*n - 9 - 3*(-1)^n + 12*(-1)^((2*n - 1 + (-1)^n)/4))/96. - _Luce ETIENNE_, Mar 21 2014
%F E.g.f.: ((2*x^3 + 9*x^2 - 3*x - 6)*cosh(x) + 6*(cos(x) + sin(x)) + (2*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48. - _Stefano Spezia_, Apr 05 2023
%e a(6) = 10 = 0 + 0 + 0 + 1 + 2 + 3 + 4.
%p A122046 := proc(n) round((2*n^3+3*n^2-8*n)/48) ; end proc: # _Mircea Merca_
%t p[n_] := p[n] = Cancel[Simplify[ (x^(n - 1)p[n - 1]p[n - 4] + p[n - 2]*p[n - 3])/p[n - 5]]]; p[ -5] = 1;p[ -4] = 1;p[ -3] = 1;p[ -2] = 1;p[ -1] = 1; Table[Exponent[p[n], x], {n, 0, 20}]
%t Accumulate[Floor[Range[0,60]^2/8]] (* or *) LinearRecurrence[{3,-3,1,1,-3,3,-1},{0,0,0,1,3,6,10},60] (* _Harvey P. Dale_, Dec 23 2019 *)
%o (Magma) [Round((2*n^3+3*n^2-8*n)/48): n in [0..60]]; // _Vincenzo Librandi_, Jun 25 2011
%o (PARI) a(n)=(2*n^3+3*n^2-8*n+3)\48 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A014125, A052762, A057077, A090294, A122047, A144678, A190718.
%Y Partial sums of A001972.
%Y Maximum Wiener index of all maximal k-degenerate graphs for k=1..6: A000292, A002623, A014125, A122046 (this sequence), A122047, A175724.
%K nonn,easy
%O 0,5
%A _Roger L. Bagula_, Sep 13 2006
%E Edited by _N. J. A. Sloane_, Sep 17 2006, Jul 11 2008, Jul 12 2008
%E More formulas and better name from _Mircea Merca_, Nov 19 2010