A261347 -id:A261347 - cp's OEIS frontend

A201126 Maximum water retention of a magic square of order n.

0, 15, 69, 192, 418, 797, 1408

Offset: 3

Hugo Pfoertner, Dec 03 2011

Determining the maximum water retention of a magic square has been the subject of the spring 2010 round of "Al Zimmermann's Programming Contests". The following description was given by Al Zimmermann: The scoring function is defined in terms of the physical characteristics of water. Simply stated, pour a gazillion units of water on top of a magic square and measure the water that doesn't run off. The cells in the magic square have heights given by their values and water cannot pass between two cells joined at a vertical edge.
Lower bounds for the next terms are a(10) >= 2267, a(11) >= 3492, a(12) >= 5185, a(13) >= 7445, a(14) >= 10397, a(15) >= 14154.
This water retention model progressed from the specific case of the magic square to a more generalized system of random levels.  A quite interesting counter-intuitive finding that a random two-level system will retain more water than a random three-level system when the size of the square is greater than 51 X 51 was discovered.  This was reported in the Physical Review Letters in 2012 and referenced in the Nature article in 2018. - Craig Knecht, Dec 01 2018

			See links for illustrations.

B. Burger, J. S. Andrade Jr. & H. J. Herrmann, A Comparison of Hydrological and Topological Water Sheds, Nature, 10586, 2018.
Harvey Heinz, Knecht Topographical squares, Summary of contest results.
Craig Knecht, Magic Square - Topographical model
Craig L. Knecht, Walter Trump, Daniel ben-Avraham, and Robert M. Ziff, Retention Capacity of Random Surfaces, Phys. Rev. Lett. 108, 045703, 2012.
Craig Knecht, Pattern comparison table.
Hugo Pfoertner, 4 X 4 Magic square retaining 15 units of water
Hugo Pfoertner, 5 X 5 Magic square retaining 69 units of water
Hugo Pfoertner, 6 X 6 Magic square retaining 192 units of water
Hugo Pfoertner, 7 X 7 Magic square retaining 418 units of water
Hugo Pfoertner, 8 X 8 Magic square retaining 797 units of water
Hugo Pfoertner, 9 X 9 Magic square retaining 1408 units of water
Wikipedia, Water retention on mathematical surfaces

Cf. A201127 (water retention of semi-magic squares), A261347 (water retention of number squares), A261798 (water retention of an associative magic square).

A303295 a(n) is the maximum water retention of a height-3 length-n number parallelogram with maximum water area.

Original entry on oeis.org

0, 20, 49, 99, 165, 247, 345, 459, 589, 735, 897, 1075, 1269, 1479, 1705, 1947, 2205, 2479, 2769, 3075, 3397, 3735, 4089, 4459, 4845, 5247, 5665, 6099, 6549, 7015, 7497, 7995, 8509, 9039, 9585, 10147, 10725, 11319, 11929, 12555

Offset: 0

Craig Knecht, Jun 15 2018

A number parallelogram contains numbers from 1 to the triangular area of the parallelogram without duplicate numbers.
This sequence applies the water retention model for mathematical surfaces to the triangular grid.
Magic polyiamond tiling is the tiling of a number shape with a single order of polyiamond. The sum of numbers in each polyiamond subspace is equal.
The height-three length-four parallelogram has an area of 24 unit triangles. The sum of the numbers from 1 to 24 is 300. Both 24 and 300 are divisible by four and six making magic polyiamond tilings possible with order four and six polyiamonds.
Five magic polyiamond tilings for a single numeric solution are noted in the link section.

Colin Barker, Table of n, a(n) for n = 0..1000
Craig Knecht, Example for the sequence.
Craig Knecht, Five magic polyiamond tilings of a single numeric solution.
Craig Knecht, Length 2 paralleogram unique dam configuration.
Craig Knecht, Magic polyiamond tiling H3 L4 Parallelogram with 99 units retained.
Craig Knecht, Water retention using a pentagonal tile.
Wikipedia, Water retention on mathematical surfaces
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A261347.

concat(0, Vec(x*(20 - 11*x + 12*x^2 - 5*x^3) / (1 - x)^3 + O(x^50))) \\ Colin Barker, Jun 15 2018

a(n) = ((4*n+7)*(4*n+2)) - (4*n+2) * (4*n+3)/2 + 4 for n > 2.
From Colin Barker, Jun 15 2018: (Start)
G.f.: x*(20 - 11*x + 12*x^2 - 5*x^3) / (1 - x)^3.
a(n) = -3 + 10*n + 8*n^2 for n>1.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 4.
(End)

A260302 Maximum water retention of a number octagon of order n.

Original entry on oeis.org

0, 0, 231, 1378, 4753, 12246, 26335, 50086, 87153, 141778, 218791, 323610, 462241, 641278, 867903, 1149886, 1495585, 1913946, 2414503, 3007378, 3703281, 4513510, 5449951, 6525078, 7751953, 9144226, 10716135, 12482506, 14458753, 16660878, 19105471, 21809710

Offset: 1

Craig Knecht, Nov 10 2015

A number octagon fills an octagon on a square grid with the smallest unique natural numbers.

The sum of the interior values for a number hexagon on a circular lattice is A079903. There are nice illustrations for this by Mathar at A257594.

			      (22 23 24)
   (37  1  2  3 25)
(36  4  5  6  7  8  26)
(35  9 10 11 12 13  27)
(34 14 15 16 17 18  28)
   (33 19 20 21 29)
      (32 31 30)
The largest values (22 - 37) form the dam with the value 22 being the spillway.

G. C. Greubel and Craig Knecht, Table of n, a(n) for n = 1..1000 (Terms 1 through 32 were computed by Craig Knecht; terms 33 through 1000 by G. C. Greubel, Nov 13 2015; term 451 = 1002105368551 corrected by _Georg Fischer_, May 24 2019)
Craig Knecht, Octagon volume calculation.
Craig Knecht, Volume retention of a number square.
Wikipedia, Water retention on mathematical surfaces
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A261347 (water retention on a number square).

[0,0] cat [(1/2)*(7*n^2-18*n+12)*(7*n^2-18*n+13): n in [3..60]]; // Vincenzo Librandi, Nov 20 2015

Table[-KroneckerDelta[n,1]  - 10*KroneckerDelta[n,2] + (1/2)*((7*n^2-18*n+12)^2+(7*n^2-18*n+12)), {n, 1, 30}] (* G. C. Greubel, Nov 13 2015 *)

concat(vector(2), Vec(-x^3*(10*x^4-49*x^3+173*x^2+223*x+231)/(x-1)^5 + O(x^100))) \\ Colin Barker, Nov 11 2015

a(n) =  (1/2)*(7*n^2 - 18*n + 12) (7*n^2 - 18*n + 13) for n > 2.
From Colin Barker, Nov 11 2015: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.
G.f.: -x^3*(10*x^4-49*x^3+173*x^2+223*x+231) / (x-1)^5. (End)

A261798 Maximum water retention of an associative magic square of order n.

Original entry on oeis.org

0, 0, 0, 15, 59, 0, 361, 704, 1247, 0

Offset: 1

Craig Knecht, Sep 01 2015

Two of the most famous magic squares are associative magic squares - the Lo Shu magic square and Dürer's magic square. Al Zimmermann's programming contest in 2010 produced the presently known maximum retention values for magic squares order 4 to 28 A201126. No concerted effort has been made to find the maximum retention for associative magic squares.
There are 4211744 different water retention patterns for a 7 x 7 square A054247 and 1.12*10^18 different order 7 associative magic squares. There is no proof that the presently stated maximum retention values greater than order 5 are actually the maximum possible retention.
a(11) >= 3226, a(12) >= 4840, a(13) >= 6972.
The Wikipedia link below shows the first attempt to classify a set of data by its water retention. Here the 48 associative order 4 magic squares are thus classified. Perhaps there might be some correlation between this surface evaluation and Mohs hardness scale.

			(16  3  2  13)
(5  10 11   8)
(9   6  7  12)
(4  15  14  1)
This is Albrecht Dürer's famous magic square in Melancholia I. Dürer put the date of its creation (1514) in the numbers in the bottom row. This square holds 5 units of water.

Craig Knecht, Order 5 associative magic square.
Craig Knecht, Order 7 associative magic square.
Craig Knecht, Order 8 associative magic square.
Craig Knecht, Order 9 associative magic square.
Craig Knecht, Order 12 associative magic square.
Johan Ofverstedt, Water Retention on Magic Squares with Constraint Based Local Search.
Wikipedia, Listing by water retention capacity. and Water retention on mathematical surfaces.

Cf. A201126 (water retention on magic squares), A201127 (water retention on semi-magic squares), A261347 (water retention on number squares).

A275359 Maximum incarceration of numbers in an n X n X n number cubes with full incarceration volumes.

Original entry on oeis.org

0, 0, 21, 292, 1566, 5664, 16375, 40716, 90552, 184576, 350649, 628500, 1072786, 1756512, 2774811, 4249084, 6331500, 9209856, 13112797, 18315396, 25145094, 33988000, 45295551, 59591532, 77479456, 99650304, 126890625, 160090996, 200254842, 248507616

Offset: 0

Craig Knecht, Jul 24 2016

The incarceration value for each cell is the highest value on the path of least resistance off the cube minus the value of the cell.  Negative values are set to zero.
This extends the idea of the 2D water retention on mathematical surfaces to the 3D cube.
A number cube contains the numbers 1 to n^3 without duplicates.
This incarceration sequence requires the smallest numbers to be placed in all possible internal cells or (n-2)^3 cells. This is not the maximum possible retention for a number cube (see link below)
Each internal cell has 6 neighbors and thus 6 initial possible pathways of escape.
A lake is a body of water that has dimensions of (n-2)x(n-2)x(n-2). All other retaining areas are called ponds. More than one lake is possible in the same cube.

			An order 3 number cube contains the numbers 1 to 27. The smallest value 1 is placed in the single central cell.  The largest possible 6 numbers 27,26,25,24,23,22 occupy the central cell in each face of the cube.  Thus the path of least resistance off the cube is through cell 22.  The total incarceration is then 22-1 = 21 units of incarceration.
2  3  4      10 27 11    14 15 16
5 23  6      24  1 25    17 22 18
7  8  9      12 26 13    19 20 21

Colin Barker, Table of n, a(n) for n = 0..1000
Craig Knecht, 7x7x7 number cube with one pond
Craig Knecht, Examples of lakes in higher order cubes
Craig Knecht, Maximum incarceration order 4 number cube
Walter Trump, Three lakes in a 6x6x6 cube
Wikipedia, Water retention on mathematical surfaces
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

A261347 (maximum retention of a number square of order n), A260302 (maximum retention of a number octagon of order n).

concat([0,0], Vec(x^2*(21+145*x-37*x^2+99*x^3+128*x^4+4*x^5)/(1-x)^7 + O(x^50))) \\ Colin Barker, Aug 01 2016

a(n) = (n^3 - 3*n^2 + 27*n - 8)/2 * (n-1)^3 \\ Charles R Greathouse IV, Aug 05 2016

a(n) = (n^3 - 3*n^2 + 27*n - 8) / 2 * (n-1)^3 for n>0.
From Colin Barker, Jul 31 2016: (Start)
a(n) = (n^6-6*n^5+39*n^4-99*n^3+108*n^2-51*n+8)/2 for n>0.
a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7) for n>7.
G.f.: x^2*(21+145*x-37*x^2+99*x^3+128*x^4+4*x^5) / (1-x)^7.
(End)

Edited and a(20)-a(29) added by Colin Barker, Aug 01 2016

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

Original entry on oeis.org

0, 0, 1, 5, 30, 74, 195, 363, 700, 1124, 1845

Offset: 1

Craig Knecht, Jan 18 2020

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.
A number square contains each of the numbers 1 to n*n exactly once.
The water retention model provides the definition of a pond. All the ponds have an area of 1 cell in the maximum pond example.
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.
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.
The cells in a given row hold identical amounts of water.
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.
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.

			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.
   0  6  0  7  0
   8  1  9  2 10
   0 11  3 12  0
  13  4 14  5 15
   0 16  0 17  0

Craig Knecht, Center of mass for the retained water
Craig Knecht, Center symmetric ponds - alternate example
Craig Knecht, Equal number of spillways and ponds
Craig Knecht, Example for the sequence
Craig Knecht, Magic squares with the maximum number of ponds
Craig L. Knecht, Walter Trump, Daniel ben-Avraham, Robert M. Ziff, Retention capacity of random surfaces, arXiv:1110.6166 [cond-mat.dis-nn], 31 Jan 2012.
Harry White, Water retention calculation and graphic display utility
Harry White, Water retention utilites.
Wikipedia, Water retention on mathematical surfaces

Cf. A201126 (water retention on magic squares), A261347 (water retention on number squares), A275359 (3 dimensional incarceration), A275339.

cp's OEIS Frontend

A201126 Maximum water retention of a magic square of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A303295 a(n) is the maximum water retention of a height-3 length-n number parallelogram with maximum water area.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A260302 Maximum water retention of a number octagon of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A261798 Maximum water retention of an associative magic square of order n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A275359 Maximum incarceration of numbers in an n X n X n number cubes with full incarceration volumes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs