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A275359 Maximum incarceration of numbers in an n X n X n number cubes with full incarceration volumes. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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).

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)

EXAMPLE

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

PROG

(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

CROSSREFS

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

Sequence in context: A230768 A230021 A101700 * A022291 A025944 A025962

Adjacent sequences:  A275356 A275357 A275358 * A275360 A275361 A275362

KEYWORD

nonn,easy

AUTHOR

Craig Knecht, Jul 24 2016

EXTENSIONS

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

STATUS

approved

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Last modified November 18 01:12 EST 2018. Contains 317279 sequences. (Running on oeis4.)