A006501 Expansion of (1+x^2) / ( (1-x)^2 * (1-x^3)^2 ). (Formerly M1091)

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

0,2

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).

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

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 *)

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).

Sequence in context: A284122 A212585 A085891 * A224814 A224810 A074633

Adjacent sequences: A006498 A006499 A006500 * A006502 A006503 A006504

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Reinhard Zumkeller, May 18 2004

Status

approved