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A331507 Water retention for an n X n number square with the maximum number of ponds using a simple filling pattern.

%I A331507 #35 Jan 24 2020 15:20:53
%S A331507 0,0,1,5,30,74,195,363,700,1124,1845
%N A331507 Water retention for an n X n number square with the maximum number of ponds using a simple filling pattern.
%C A331507 The number placement starts with the lowest available number and proceeds from top left to bottom right in two separate passes. The first pass fills in the ponds. The second pass fills in the barrier cells surrounding the ponds.
%C A331507 A number square contains each of the numbers 1 to n*n exactly once.
%C A331507 The water retention model provides the definition of a pond. All the ponds have an area of 1 cell in the maximum pond example.
%C A331507 The immediate environment of a 1-cell pond requires four larger surrounding cells. The water retention model requires the macro environment of possible surrounding cells to be lower than the border of the 1-cell-area pond.
%C A331507 For even-ordered squares one of the main diagonals is made up of ponds. For odd-ordered squares both diagonals are made up of ponds.
%C A331507 The cells in a given row hold identical amounts of water.
%C A331507 A listing of the C code that calculates the water retention is given. The program gives a graphic output where the area of the ponds is color coded. Additional 3D graphics and other water retention utilites are available on Harry White's web page noted below.
%C A331507 The water retention model functions in three dimensions as noted in the crossrefs. The physical interpretation in three dimensions is not straightforward and the term "incarceration" of numbers is introduced.
%H A331507 Craig Knecht, <a href="/A331507/a331507_2.png">Center of mass for the retained water</a>
%H A331507 Craig Knecht, <a href="/A331507/a331507_4.png">Center symmetric ponds - alternate example</a>
%H A331507 Craig Knecht, <a href="/A331507/a331507.png">Equal number of spillways and ponds</a>
%H A331507 Craig Knecht, <a href="/A331507/a331507_1.png">Example for the sequence</a>
%H A331507 Craig Knecht, <a href="/A331507/a331507_3.png">Magic squares with the maximum number of ponds</a>
%H A331507 Craig L. Knecht, Walter Trump, Daniel ben-Avraham, Robert M. Ziff, <a href="https://arxiv.org/abs/1110.6166">Retention capacity of random surfaces</a>, arXiv:1110.6166 [cond-mat.dis-nn], 31 Jan 2012.
%H A331507 Harry White, <a href="/A331507/a331507.txt">Water retention calculation and graphic display utility</a>
%H A331507 Harry White, <a href="http://budshaw.ca/Download.html">Water retention utilites</a>.
%H A331507 Wikipedia, <a href="http://en.wikipedia.org/wiki/Water retention on mathematical surfaces">Water retention on mathematical surfaces</a>
%e A331507 Order 5 square retaining 30 units of water. The positions of the remaining numbers that do not occupy the pond cells or their immediate borders are irrelevant and a zero is placed in these positions.
%e A331507    0  6  0  7  0
%e A331507    8  1  9  2 10
%e A331507    0 11  3 12  0
%e A331507   13  4 14  5 15
%e A331507    0 16  0 17  0
%Y A331507 Cf. A201126 (water retention on magic squares), A261347 (water retention on number squares), A275359 (3 dimensional incarceration), A275339.
%K A331507 nonn,more
%O A331507 1,4
%A A331507 _Craig Knecht_, Jan 18 2020