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%I M1419 N0556 #31 Feb 20 2022 07:15:33
%S 1,2,5,12,32,94,289,910,2934,9686,32540,110780,381676,1328980,4669367,
%T 16535154,58965214,211591218,763535450,2769176514,10089240974,
%U 36912710568,135565151486,499619269774,1847267563742,6850369296298
%N Balancing weights on the integer line.
%C 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
%C Is this a shifted version of A076822? - _Vladimir Reshetnikov_, Oct 06 2016
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. E. Odeh and E. J. Cockayne, <a href="http://dx.doi.org/10.1016/S0021-9800(69)80047-5">Balancing weights on the integer line</a>, J. Combin. Theory, 7 (1969), 130-135.
%F 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
%t (* 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 *)
%Y Cf. A047997, A076822, A188181 (columns 1, 2).
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Henry Bottomley_, Aug 09 2002
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