A201126 -id:A201126 - cp's OEIS frontend

A101089 Second partial sums of fourth powers (A000583).

1, 18, 116, 470, 1449, 3724, 8400, 17172, 32505, 57838, 97812, 158522, 247793, 375480, 553792, 797640, 1125009, 1557354, 2120020, 2842686, 3759833, 4911236, 6342480, 8105500, 10259145, 12869766, 16011828, 19768546, 24232545, 29506544

Offset: 1

Cecilia Rossiter, Dec 14 2004

a(n) is the n-th antidiagonal sum of the convolution array A213553. - Clark Kimberling, Jun 17 2012
a(n-1)/n^5 is the "retention" of water on a 3 X 3 random surface of n levels - see Knecht et al., 2012, Schrenk et al., 2014. - Robert M. Ziff, Mar 08 2014
The general formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) is the m-th Faulhaber’s polynomial. - Luciano Ancora, Jan 26 2015

			a(7) = 8400 = 1*(8-1)^4 + 2*(8-2)^4 + 3*(8-3)^4 + 4*(8-4)^4 + 5*(8-5)^4 + 6*(8-6)^4 + 7*(8-7)^4. - _Bruno Berselli_, Jan 31 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Luciano Ancora, Recurrence relation for the second partial sums of m-th powers.
Luciano Ancora, Second partial sums of the m-th powers.
Craig L. Knecht, Walter Trump, Daniel ben-Avraham and Robert M. Ziff, Retention Capacity of Random Surfaces, Phys. Rev. Lett., Vol. 108 (2012), 045703.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev., Vol. 19, No. 1 (1977), pp. 90-99. MR0428678 (55 #1698). See Table 1. - _N. J. A. Sloane_, Mar 23 2014
Claudio de J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq., Vol. 16 (2013), Article 13.5.7.
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]
Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Cached copy, May 15 2013]
K. J. Schrenk, N. A. M. Araújo, R. M. Ziff and H. J. Herrmann Retention Capacity of Correlated Surfaces, arXiv:1403.2082 [cond-mat.stat-mech], 2014.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Partial sums of A000538.

Cf. A000583, A002415, A101090, A201126, A213553.

List([1..40], n-> (n+1)^2*(2*(n+1)^4-5*(n+1)^2+3)/60); # G. C. Greubel, Jul 31 2019

[(1/60)*n*(n+1)^2*(n+2)*(2*n*(n+2)-1): n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

f:=n->(2*n^6-5*n^4+3*n^2)/60;
[seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 23 2014

a[n_] := n(n+1)^2(n+2)(2n(n+2) -1)/60; Table[a[n], {n, 40}]
CoefficientList[Series[(1+x)*(1+10*x+x^2)/(1-x)^7, {x,0,40}], x] (* Vincenzo Librandi, Mar 24 2014 *)
Nest[Accumulate[#]&,Range[30]^4,2] (* Harvey P. Dale, Aug 13 2024 *)

a(n)=n*(n+1)^2*(n+2)*(2*n*(n+2)-1)/60 \\ Charles R Greathouse IV, Mar 18 2014

[n*(n+1)^2*(n+2)*(2*n*(n+2)-1)/60 for n in range(1,40)] # Danny Rorabaugh, Apr 20 2015

a(n) = (1/60)*n*(n+1)^2*(n+2)*(2*n*(n+2)-1).
G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^7. - Colin Barker, Apr 04 2012
a(n) = Sum_{i=1..n} i*(n+1-i)^4, by the definition. - Bruno Berselli, Jan 31 2014
a(n) = 2*a(n-1) - a(n-2) + n^4. - Luciano Ancora, Jan 08 2015
Sum_{n>=1} 1/a(n) = 85/3 + 10*Pi^2/3 - 20*sqrt(2/3)*Pi*cot(sqrt(3/2)*Pi). - Amiram Eldar, Jan 26 2022
a(n) = (1/2)*Sum_{1 <= i, j <= n+1} (i - j)^4 - Peter Bala, Jun 11 2024

Edited by Ralf Stephan, Dec 16 2004

A261347 Maximum water retention of a number square of order n.

Original entry on oeis.org

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

Offset: 1

Craig Knecht, Aug 15 2015

nonn

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.)

			(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.

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
Wikipedia, Retention with prime numbers.

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

A201127 Maximum water retention of a semi-magic square of order n.

Original entry on oeis.org

4, 22, 78, 199, 424, 814, 1410

Offset: 3

Hugo Pfoertner, Dec 03 2011

The same rules as for A201126 apply, but with the magic conditions for both diagonals of the number square removed.

a(10) >= 2280. - Hugo Pfoertner, May 19 2012

			(7 6 2)
(5 1 9)
(3 8 4)
is a semi-magic square. The mid-side bricks with heights 6, 5, 9, 8 form a wall around the central hole with bottom height 1. Water poured upon the square will fill the central pond until overflowing via the left brick of height 5. Thus 4 units of water will be retained.

Hugo Pfoertner, 3 X 3 Semi-magic square retaining 4 units of water
Hugo Pfoertner, 4 X 4 Semi-magic square retaining 22 units of water
Walter Trump, 5 X 5 Semi-magic square retaining 78 units of water
Hugo Pfoertner, 6 X 6 Semi-magic square retaining 199 units of water
Hugo Pfoertner, 7 X 7 Semi-magic square retaining 424 units of water
Hugo Pfoertner, 8 X 8 Semi-magic square retaining 814 units of water
Hugo Pfoertner, 9 X 9 Semi-magic square retaining 1410 units of water
Wikipedia, Water retention on mathematical surfaces

Cf. A201126 (water retention of magic squares).

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).

A285766 Maximum spillway height for a zero or one bend minimal area lake in a number square.

Original entry on oeis.org

0, 0, 6, 10, 15, 22, 31, 42, 55, 70, 87, 106, 127, 150, 175, 202, 231, 262, 295, 330, 367, 406, 447, 490, 535, 582, 631, 682, 735, 790, 847, 906, 967, 1030, 1095, 1162, 1231, 1302, 1375, 1450, 1527, 1606, 1687, 1770, 1855, 1942, 2031, 2122, 2215, 2310, 2407

Offset: 0

Craig Knecht, May 04 2017

nonn

The water retention model for mathematical surfaces led to definitions for a lake and a pond.  These lakes and ponds divide the square up in interesting ways. This sequence looks at the spillway heights in zero or one bend minimal area lakes.
A lake has dimensions of (n-2) X (n-2) when the square is n X n.  All other water retaining areas are ponds.
A number square contains the numbers 1 to n^2 without repeats.
The larger terms are a(n)= n^2+6 or A114949.

			For the 4 X 4 square a example of a smallest lake is shown. The values 1,2,3 form the lake. The pathway of least resistance off the square is the spillway value 10.
   ( 4  16  15   5)
   (10   1   2  14)
   ( 6  11   3  13)
   ( 7   8  12   9)

Craig Knecht, Minimal lake types in a 7x7 square.
Craig Knecht, Minimal lake area in a square
Wikipedia, Water retention on mathematical surfaces

Cf. A054247, A201126, A268311.

Conjectures from Colin Barker, May 07 2017: (Start)
G.f.: x^2*(6 - 8*x + 3*x^2 + x^3) / (1 - x)^3.
a(n) = 7 - 2*n + n^2 for n>2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>5.
(End)

A286430 Least volume of water to surround the largest possible island in a number square.

Original entry on oeis.org

0, 0, 0, 0, 36, 78, 136, 210, 300, 406, 528, 666, 820, 990, 1176, 1378, 1596, 1830, 2080, 2346, 2628, 2926, 3240, 3570, 3916, 4278, 4656, 5050, 5460, 5886, 6328, 6786, 7260

Offset: 0

Craig Knecht, May 09 2017

nonn

The water retention model for mathematical surfaces showed that a random two-level system will contain more water than a random 3-level system when the size of the square is > 52 X 52. It has also been the subject of Zimmermann's programming contest in 2010 and a Wikipedia page as noted below. The number square is a simple environment in which to explore the interaction of volumes, heights, and areas of lakes, ponds, islands, and spillways in the square.
A number square contains the numbers for 1 to n^2 without repeats in an n X n square.
This sequence is 4*A000217 for a(n) > 8.

			For this 5 X 5 square the numbers 1 to 25 are used without repeats. The values 1 through 8 form the moat.  The spillway value is 9.  The volume of water retained is 36 units.
( 24 23 22 21 20)
( 18  1  2  3 19)
( 17  8 25  4  9)
( 16  7  6  5 15)
( 14 13 12 11 10)

Craig Knecht, 3D graphic.
Craig Knecht, Least volume of water to surround the largest island in a number square.
Craig Knecht, Number range for each cell type.
Wikipedia, Water retention on mathematical surfaces

Cf. A054247, A201126, A268311.

Conjectures from Colin Barker, Jan 20 2018: (Start)
G.f.: 2*x^4*(18 - 15*x + 5*x^2) / (1 - x)^3.
a(n) = 28 - 30*n + 8*n^2 for n>3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2.
(End)

A286429 Highest elevation of an island above sea level in a number square.

Original entry on oeis.org

0, 0, 0, 0, 16, 27, 40, 55, 72, 91, 112, 135, 160, 187, 216, 247, 280, 315, 352, 391, 432, 475, 520, 567, 616, 667, 720, 775, 832, 891, 952, 1015, 1080, 1147, 1216, 1287, 1360, 1435, 1512, 1591, 1672, 1755, 1840, 1927, 2016, 2107, 2200, 2295, 2392

Offset: 0

Craig Knecht, May 09 2017

nonn

The water retention model for mathematical surfaces has previously looked at lakes and ponds. This sequence looks at the maximum possible height of an island above water level in a number square.
The smallest possible water elevation will always be composed of an eight-cell lake or pond with a spillway value of nine. This moat is not centered in a(n) > 5 but has the square's edge as one of its borders.
A number square contains the numbers 1 to n^2 without repeats.
The larger terms in this sequence are a(n) = n*(n+6) or A028560.

			For the 6 X 6 number square the largest value is 36 which is assigned to the single-cell island.
I only include the pertinent moat, spillway, and island values for the 6 X 6 example.
( 1   2   3    )
( 8  36   4  9 )
( 7   6   5    )

Craig Knecht, Maximum island elevation above sea level in a number square
Wikipedia, Water retention on mathematical surfaces

Cf. A054247, A201126.

Conjectures from Colin Barker, May 09 2017: (Start)
G.f.: x^4*(16 - 21*x + 7*x^2) / (1 - x)^3.
a(n) = n^2 + 2*n - 8 for n>3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>6.
(End)

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.

A350413 Total capacity of unrestricted compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 1, 6, 24, 79, 233, 640, 1674, 4224, 10370, 24912, 58800, 136767, 314201, 714209, 1608604, 3594007, 7973108, 17576844, 38531060, 84040238, 182467033, 394537387, 849885436, 1824495630, 3904453264, 8331493468, 17730810536, 37641301271, 79727649682, 168512283733

Offset: 1

Michel Marcus, Dec 29 2021

nonn

Aubrey Blecher, Charlotte Brennan, and Arnold Knopfmacher, The water capacity of integer compositions, Online Journal of Analytic Combinatorics, Issue 13, 2018, #6.

Cf. A201126, A285766.

v(z, N) = (-2 + 2*N + 5*z - 5*N*z - 4*z^2 + 3*N*z^2)/(-1 + 2*z)^2 + (1 - z)*(-3 + 4*z)/(1 - 2*z)*sum(r=2, N, 1/(1 - 2*z + z^r), O(z*z^N)) + sum(r=2, N, (1 - z)^2/(1 - 2*z + z^r)^2, O(z*z^N));
lista(nn) = Vec(v(z,nn), -nn);

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A101089 Second partial sums of fourth powers (A000583).

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A261347 Maximum water retention of a number square of order n.

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A201127 Maximum water retention of a semi-magic square of order n.

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

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A285766 Maximum spillway height for a zero or one bend minimal area lake in a number square.

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A286430 Least volume of water to surround the largest possible island in a number square.

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A286429 Highest elevation of an island above sea level in a number square.

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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.

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A350413 Total capacity of unrestricted compositions of n.

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