A002838 Balancing weights on the integer line. (Formerly M1419 N0556)

1, 2, 5, 12, 32, 94, 289, 910, 2934, 9686, 32540, 110780, 381676, 1328980, 4669367, 16535154, 58965214, 211591218, 763535450, 2769176514, 10089240974, 36912710568, 135565151486, 499619269774, 1847267563742, 6850369296298

Offset

1,2

Comments

Also number of partitions of n(n+1)/2 into up to n parts each no greater than n+1, partitions of n(n+3)/2 into exactly n parts each no greater than n+2 and partitions of n(n+1) into exactly n distinct parts each no greater than 2n+1, thus providing balancing solutions for n weights in distinct integer positions on [ -n,n] with a pivot at 0. - Henry Bottomley, Aug 09 2002

Is this a shifted version of A076822? - Vladimir Reshetnikov, Oct 06 2016

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

Table of n, a(n) for n=1..26.

R. E. Odeh and E. J. Cockayne, Balancing weights on the integer line, J. Combin. Theory, 7 (1969), 130-135.

Formula

a(n) = A047997(n, n) = A067059(n, n+1). a(n) tends towards (sqrt(12)/Pi)*4^n/n^2 and something like (sqrt(12)/Pi)*4^n/(n^2+1.85*n+0.8) seems to give an even closer approximation. - Henry Bottomley, Aug 09 2002

Mathematica

(* This program is not convenient for large values of n *) a[n_] := Length[ IntegerPartitions[n*(n+1)/2, n, Range[n+1]]]; Table[ Print[{n, an = a[n]}]; an, {n, 1, 16}] (* Jean-François Alcover, Jan 02 2013 *)

Crossrefs

Cf. A047997, A076822, A188181 (columns 1, 2).

Sequence in context: A148283 A218781 A363682 * A076822 A143657 A014326

Adjacent sequences: A002835 A002836 A002837 * A002839 A002840 A002841

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Henry Bottomley, Aug 09 2002

Status

approved