2015 update - Cylindrical Models

 

Cylinder.jpg

 

Pi day 3-14-15 brought about the idea of a cylindrical model for water retention

 

https://en.wikipedia.org/wiki/User:Knecht03/sandbox

Miguel Angel Amela

Miguel’s brain was in fine form working on the cylindrical model for water retention. His illustrations from a flattened Pascal’s triangle (concatenation of the rows) are shown below.

 

rabbit.png

 

Top of Form

Bottom of Form

A258445

Irregular triangle related to Pascal's triangle.

5

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 4, 4, 6, 4, 4, 1, 1, 1, 1, 5, 5, 10, 10, 10, 5, 5, 1, 1, 1, 1, 6, 6, 15, 15, 20, 15, 15, 6, 6, 1, 1, 1, 1, 7, 7, 21, 21, 35, 35, 35, 21, 21, 7, 7, 1, 1, 1, 1, 8, 8, 28, 28, 56, 56, 70, 56, 56, 28, 28, 8, 8, 1, 1, 1, 1, 9, 9, 36, 36, 84, 84, 126, 126, 126, 84, 84, 36, 36, 9, 9, 1, 1 (list; graph; refs; listen;history; text; internal format)

OFFSET

1,7

COMMENTS

The sequence of row lengths of this irregular triangle T(n, k) isA005408(n-1) = 2*n -1.

This array represents the height of water retention between a collection of cylinders whose height and arrangement are specified by Pascal's triangle.

The row sums for this retention are A164991.

Each term is the minimum of 3 terms of the Pascal's triangle: 2 terms below and 1 above when k is odd, and 2 terms above and 1 below when k is even. - Michel Marcus, Jun 11 2015

LINKS

Table of n, a(n) for n=1..100.

Miguel Angel AmelaFractal Antenna

Miguel Angel AmelaPascal Wave

Craig Knecht, Pascal's Neighborhood

Craig Knecht, Pascal Surface

Craig Knecht, Pascal Cylinders

Wikipedia, Water Retention on Mathematical Surfaces

FORMULA

T(n, 2*m) = Min(P(n-1, m-1), P(n-1, m), P(n, m)) with P(n, k) = A00731(n, k) = binomial(n, k), for m = 1, 2, ..., n-1, and

T(n, 2*m-1) = Min(P(n-1, m-1), P(n, m-1), P(n, m)) for m = 1, 2, ..., n. See the program by Michel Marcus. - Wolfdieter Lang, Jun 27 2015

EXAMPLE

The irregular triangle T(n, k) starts:

n\k 1 2 3 4  5  6  7  8  9 10 11 12 13 14 15 16 17

1:  1

2:  1 1 1

3:  1 1 2 1  1

4:  1 1 3 3  3  1  1

5:  1 1 4 4  6  4  4  1  1

6:  1 1 5 5 10 10 10  5  5  1  1

7:  1 1 6 6 15 15 20 15 15  6  6  1  1

8:  1 1 7 7 21 21 35 35 35 21 21  7  7  1  1

9:  1 1 8 8 28 28 56 56 70 56 56 28 28  8  8  1  1

... Reformatted. - Wolfdieter Lang, Jun 26 2015

PROG

(PARI) tabf(nn) = {for (n=1, nn, for (k=1, 2*n-1, kk = (k+1)\2; if (k%2, v = min(binomial(n-1, kk-1), min(binomial(n, kk-1), binomial(n, kk))), v = min(binomial(n, kk), min(binomial(n-1, kk-1), binomial(n-1, kk)))); print1(v, ", "); ); print(); ); } \\ Michel Marcus, Jun 16 2015

CROSSREFS

Cf. A007318 (Pascal's triangle), A164991.

Sequence in context: A219924 A226444 A196929 * A129179 A120621 A201080

Adjacent sequences:  A258442 A258443 A258444 * A258446 A258449 A258450

KEYWORD

nonn,tabf,easy

AUTHOR

Craig Knecht, May 30 2015

STATUS

approved

 

The water height between cylinders is used to construct a derived triangle

Gift.GIF

 

 

 

Pascal.png

 

cow.png

 

This fellow wandered into the yard and got himself posted on the magic square history update.

Magic Square History.jpg

Magic square history starts with a 3x3 magic square found on a turtle's back. The turtle got a boost up from the computer chip. Lego blocks indicate the present ability to explore the physical properties of magic squares. The turtle will not claim Alex Honnold status until he free solos the 8A(+)(v12) order 6 magic square enumeration.

 

Peano Magic Square.jpg

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Top of Form

Bottom of Form

 

A261798

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

0

0, 0, 0, 15, 59, 0, 361, 704, 1247, 0 (list; graph; refs; listen; history; published version; edit; text; internal format)

OFFSET

1,4

COMMENTS

Two of the most famous magic squares are associative magic squares - the Lo Shu magic square and Dürer's magic square. Al Zimmermann's programing 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 squareA054247 and 1.12 E18 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 associative magic square 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.

LINKS

Table of n, a(n) for n=1..10.

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, Associative magic square

Wikipedia, Magic square construction

Wikipedia, Water retention on mathematical surfaces

EXAMPLE

(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 Melencolia I. Dürer put the date of its creation (1514) in the numbers in the bottom row. This square holds 5 units of water.

CROSSREFS

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

Sequence in context: A261419 A183942 A012691 A020187 A022287 A223344A206238

Adjacent sequences:  A261785 A261790 A261796 A261797 A261799 A261800A261801

KEYWORD

nonn,more,changed

AUTHOR

Craig Knecht, Sep 01 2015

STATUS

proposed

 

A261347

Maximum water retention of a number square of order n.

0

0, 0, 5, 26, 84, 222, 488, 946, 1664, 2723, 4227, 6277, 8993, 12514, 16976, 22538, 29364, 37649, 47563, 59321, 73149, 89254, 107892, 129308, 153764, 181547, 212931, 248223, 287747, 331780 (list; graph; refs; listen; history; published version; edit; text; internal format)

OFFSET

1,3

COMMENTS

A number square is an arrangement of numbers from 1 to n*n in an n X n matrix with each number used only once.

The number square was used in 2009 as a stepping stone in solving the problem of finding the maximum water retention for magic squares.

In June 2009, Walter Trump wrote a program that calculates the maximum water retention in number squares up to 250 X 250.

The retention patterns for orders 3, 4, 8 and 11 show perfect symmetry.

For orders 5, 7, 30 and 58, more than one pattern gives maximum retention. (For order 7, there are 3 patterns that give maximum retention.)

LINKS

Table of n, a(n) for n=1..30.

Craig Knecht, Maximum retention 5 X 5 number square.

Craig Knecht, Maximum retention 6 X 6 number square.

Craig Knecht, Maximum retention 7 X 7 number square.

Craig Knecht, Maximum retention 8 X 8 number square.

Craig Knecht, Maximum retention 9 X 9 number square.

Craig Knecht, Order 7 - three patterns for maximum retention.

Craig Knecht, Order 30 - two patterns for maximum retention.

Craig Knecht, Pattern comparison table.

Wikipedia, Water retention on mathematical surfaces

EXAMPLE

(2 6 3)

(7 1 8)

(4 9 5)

  The values 6,7,8,9 form the dam with the value 6 being the spillway. 5 units of water are retained above the central cell. The boundaries of the system are open and allow water to flow out.

CROSSREFS

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

Sequence in context: A145013 A096943 A166810 A210367 A079909 A047669A002316

Adjacent sequences:  A261342 A261343 A261345 A261346 A261350 A261351A261352

KEYWORD

nonn,changed

AUTHOR

Craig Knecht, Aug 15 2015

 

Magic Square 2015.jpeg

This most perfect magic square begins with the 2015 date.

 

https://en.wikipedia.org/wiki/Most-perfect_magic_square