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

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

Author

Craig Knecht, Jul 24 2016

Keywords

• nonn
• easy

Comments

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.

Examples

			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
		

Links

• 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).

Crossrefs

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

Programs

PARI

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

PARI

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

Formula

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)

Extensions

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