A006501 Expansion of (1+x^2) / ( (1-x)^2 * (1-x^3)^2 ).

1, 2, 4, 8, 12, 18, 27, 36, 48, 64, 80, 100, 125, 150, 180, 216, 252, 294, 343, 392, 448, 512, 576, 648, 729, 810, 900, 1000, 1100, 1210, 1331, 1452, 1584, 1728, 1872, 2028, 2197, 2366, 2548, 2744, 2940, 3150, 3375, 3600, 3840, 4096, 4352, 4624, 4913, 5202

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

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

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• G. E. Bergum and V. E. Hoggatt, Jr., A combinatorial problem involving recursive sequences and tridiagonal matrices, Fib. Quart., 16 (1978), 113-118.
• Dhruv Mubayi, Counting substructures II: Hypergraphs, Combinatorica 33 (2013), no. 5, 591--612. MR3132928.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Crossrefs

Cf. A000578, A002117, A144677, A210433.

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).

Programs

Maple

A006501:=(1+z**2)/(z**2+z+1)**2/(z-1)**4; # Simon Plouffe in his 1992 dissertation

Mathematica

CoefficientList[Series[(1+x^2)/(1-x)^2 /(1-x^3)^2,{x,0,50}],x] (* Vincenzo Librandi, Jun 16 2012 *)

Formula

a(n) = [(n+3)/3] * [(n+4)/3] * [(n+5)/3]. - Reinhard Zumkeller, May 18 2004

a(n-3) = Sum_{k=0..n} [k/3]*[(k+1)/3]. - Mitch Harris, Dec 02 2004

Conjecture: a(n) = A144677(n) + A144677(n-2). - R. J. Mathar, Mar 15 2011

Sum_{n>=0} 1/a(n) = 1 + zeta(3). - Amiram Eldar, Jan 10 2023

a(3*m) = (m+1)^3 (A000578). - Bernard Schott, Feb 22 2023

Extensions

More terms from Reinhard Zumkeller, May 18 2004