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%I #35 Apr 04 2023 07:45:57
%S 0,1,4,12,31,70,141,259,442,711,1090,1606,2289,3172,4291,5685,7396,
%T 9469,11952,14896,18355,22386,27049,32407,38526,45475,53326,62154,
%U 72037,83056,95295,108841,123784,140217,158236,177940,199431,222814,248197,275691,305410
%N For n weights, number of combinations when limited to two weights per pan.
%C For 4 weights, 1, 3, 8, 23 works for values up to 28. For 5 weights, 10, 12, 13, 17, 51 works up to 56. The lowest set of n weights with f(n) distinct values is still unknown at this time.
%C Binomial transform of the sequence (0, 1, 2, 3, 3, 0, 0, 0, ....). - _Paul Barry_, Sep 05 2005
%D Discovered by Tom Turrittin and Ed Pegg Jr.
%H Vincenzo Librandi, <a href="/A037255/b037255.txt">Table of n, a(n) for n = 0..1000</a>
%H Michal Opler, Pavel Valtr, and Tung Anh Vu, <a href="https://dccg.upc.edu/eurocg23/wp-content/uploads/2023/03/Session-7B-Talk-1.pdf">On the Arrangement of Hyperplanes Determined by n Points</a>, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
%H Ed Pegg Jr., <a href="http://mathpuzzle.com/Solution.htm">Commentary on weekly puzzles</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = (n^4 - 2*n^3 + 7*n^2 + 2*n) / 8.
%F G.f.: -x*(x^3+2*x^2-x+1) / (x-1)^5. - _Colin Barker_, Apr 16 2013 [corrected by _Georg Fischer_, May 11 2019]
%t CoefficientList[Series[- x (x^3 + 2 x^2 - x + 1)/(x - 1)^5, {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 21 2013 *)
%t LinearRecurrence[{5,-10,10,-5,1},{0,1,4,12,31},50] (* _Harvey P. Dale_, Sep 03 2015 *)
%o (Magma) [(n^4-2*n^3+7*n^2+2*n)/8: n in [0..40]]; // _Vincenzo Librandi_, Oct 21 2013
%o (Python)
%o from __future__ import division
%o A037255_list = [n*(n*(n*(n - 2) + 7) + 2)//8 for n in range(10**3)] # _Chai Wah Wu_, Jan 22 2015
%Y Cf. A038523.
%K easy,nonn
%O 0,3
%A _Ed Pegg Jr_
%E More terms from _Vincenzo Librandi_, Oct 21 2013
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