%I M1091 #69 Feb 24 2023 04:54:59

%S 1,2,4,8,12,18,27,36,48,64,80,100,125,150,180,216,252,294,343,392,448,

%T 512,576,648,729,810,900,1000,1100,1210,1331,1452,1584,1728,1872,2028,

%U 2197,2366,2548,2744,2940,3150,3375,3600,3840,4096,4352,4624,4913,5202

%N Expansion of (1+x^2) / ( (1-x)^2 * (1-x^3)^2 ).

%C a(n+3) = maximal product of three numbers with sum n: a(n) = max(r*s*t), n = r+s+t. - _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 10 2003

%C It appears that k is a term of the sequence if and only if k is a positive integer such that floor(v) * ceiling(v) * round(v) = k, where v = k^(1/3). - _John W. Layman_, Mar 21 2012

%C The sequence floor(n/3)*floor((n+1)/3)*floor((n+2)/3) is essentially the same: 0, 0, 0, 1, 2, 4, 8, 12, 18, 27, 36, 48, 64, 80, 100, 125, 150, 180, 216, 252, ... - _N. J. A. Sloane_, Dec 27 2013

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A006501/b006501.txt">Table of n, a(n) for n = 0..1000</a>

%H G. E. Bergum and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/16-2/bergum.pdf">A combinatorial problem involving recursive sequences and tridiagonal matrices</a>, Fib. Quart., 16 (1978), 113-118.

%H Dhruv Mubayi, <a href="http://dx.doi.org/10.100/s00493-013-2638-2">Counting substructures II: Hypergraphs</a>, Combinatorica 33 (2013), no. 5, 591--612. MR3132928.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,2,-4,2,-1,2,-1).

%F a(n) = [(n+3)/3] * [(n+4)/3] * [(n+5)/3]. - _Reinhard Zumkeller_, May 18 2004

%F a(n-3) = Sum_{k=0..n} [k/3]*[(k+1)/3]. - _Mitch Harris_, Dec 02 2004

%F Conjecture: a(n) = A144677(n) + A144677(n-2). - _R. J. Mathar_, Mar 15 2011

%F Sum_{n>=0} 1/a(n) = 1 + zeta(3). - _Amiram Eldar_, Jan 10 2023

%F a(3*m) = (m+1)^3 (A000578). - _Bernard Schott_, Feb 22 2023

%p A006501:=(1+z**2)/(z**2+z+1)**2/(z-1)**4; # _Simon Plouffe_ in his 1992 dissertation

%t CoefficientList[Series[(1+x^2)/(1-x)^2 /(1-x^3)^2,{x,0,50}],x] (* _Vincenzo Librandi_, Jun 16 2012 *)

%Y Cf. A000578, A002117, A144677, A210433.

%Y Maximal product of k positive integers with sum n, for k = 2..10: A002620 (k=2), this sequence (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Reinhard Zumkeller_, May 18 2004