A068748 -id:A068748 - cp's OEIS frontend

A068744 Number of potential flows in 3 X 3 array with integer velocities in -n..n, i.e., number of 3 X 3 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

1, 1665, 87825, 1253329, 9230193, 45642289, 172989921, 542131425, 1473095713, 3582226465, 7970825457, 16492629297, 32119620625, 59427841617, 105227044417, 179360179905, 295700892993, 473379359425, 738268965841

Offset: 0

R. H. Hardin, Feb 27 2002

nonn

Let y = 2*n - 1; Then apparently a(n) = y^2*(529*y^6 + 910*y^4 + 721*y^2 + 360)/2520. See A068745 (4 X 4) and A063496 (2 X 2), which is y*(2*y^2 + 1)/3 under the same transformation. Suggests total degree N X N-1, with a factor y or y^2 to make the remaining polynomial even. - R. H. Hardin, Jan 02 2007

2 X 2 A063496, 4 X 4 A068745, 5 X 5 A068746, 6 X 6 A068747, by velocity limit 1..14 A068748-A068761, solenoidal flows A068722-A068738.

Empirical G.f.: -(x^8+1656*x^7 +72876*x^6 +522760*x^5 +972198*x^4 +522760*x^3 +72876*x^2 +1656*x +1)/(x-1)^9. [Colin Barker, Jul 31 2012]

Empirical: 315*a(n) = (4232*n^6 +12696*n^5 +17690*n^4 +14220*n^3 +7058*n^2 +2064*n +315) *(1+2*n)^2. - R. J. Mathar, Nov 09 2018

A068747 Number of potential flows in 6 X 6 array with integer velocities in -n..n, i.e., number of 6 X 6 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 12961320464027, 379827509377146575439, 40918963843906494595507077

Offset: 0

R. H. Hardin, Feb 27 2002

nonn

Cf. 2 X 2 A063496, 3 X 3 A068744, 4 X 4 A068745, 5 X 5 A068746, by velocity limit 1..14 A068748-A068761, solenoidal flows A068722-A068738.

A068745 Number of potential flows in 4 X 4 array with integer velocities in -n..n, i.e., number of 4 X 4 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 690437, 1133641543, 164185416899, 6913624013061, 138190481342321, 1678843050246451, 14285299502131463, 93044501704039945, 492225938556374973, 2204710243834695807, 8617480381892283531

Offset: 0

R. H. Hardin, Feb 27 2002

nonn

2 X 2 A063496, 3 X 3 A068744, 5 X 5 A068746, 6 X 6 A068747, by velocity limit 1..14 A068748-A068761, solenoidal flows A068722-A068738.

Cf. 4 X 4 this sequence (degree 4*4-1) with factor 2n-1 ; 3 X 3 A068744 (degree 3*3-1) with factor (2n-1)^2 ; 2 X 2 A063496 (degree 2*2-1) with factor 2n-1.

Let y = 2*n - 1; it appears that a(n) = y*(2623243666*y^14 + 9598591135*y^12 + 17180805187*y^10 + 20342655905*y^8 + 17636121503*y^6 + 10907793260*y^4 + 3135618144*y^2 + 304819200)/81729648000. - R. H. Hardin, Jan 01 2007

A068746 Number of potential flows in 5 X 5 array with integer velocities in -n..n, i.e., number of 5 X 5 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 1366395515, 184422574177355, 523957519578572209, 207345516734034667209, 24953087551680958151267, 1354915464537160758459123

Offset: 0

R. H. Hardin, Feb 27 2002

nonn

2 X 2 A063496, 3 X 3 A068744, 4 X 4 A068745, 6 X 6 A068747, by velocity limit 1..14 A068748-A068761, solenoidal flows A068722-A068738.

A222169 T(n,k)=Number of nXk 0..4 arrays with entries increasing mod 5 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

Original entry on oeis.org

1, 3, 3, 9, 19, 9, 27, 121, 121, 27, 81, 771, 1665, 771, 81, 243, 4913, 22979, 22979, 4913, 243, 729, 31307, 317259, 690437, 317259, 31307, 729, 2187, 199497, 4380445, 20780181, 20780181, 4380445, 199497, 2187, 6561, 1271251, 60481881, 625649047

Offset: 1

R. H. Hardin Feb 10 2013

Table starts
......1..........3..............9.................27.....................81
......3.........19............121................771...................4913
......9........121...........1665..............22979.................317259
.....27........771..........22979.............690437...............20780181
.....81.......4913.........317259...........20780181.............1366395515
....243......31307........4380445..........625649047............89948464453
....729.....199497.......60481881........18838482047..........5923189816253
...2187....1271251......835088891.......567241901289........390086038882651
...6561....8100769....11530288395.....17080173559277......25690815631493191
..19683...51620379...159201677509....514300085627023....1691995329032459285
..59049..328939577..2198138788809..15486061794514775..111434983000652039093
.177147.2096095523.30350271502115.466299978310573033.7339124863989795685471

			Some solutions for n=3 k=4
..0..1..3..3....0..2..3..3....0..1..2..4....0..1..3..4....0..0..0..1
..2..3..0..0....0..2..4..4....1..2..3..4....1..3..4..1....0..2..2..2
..4..0..0..0....2..4..4..0....1..2..3..4....3..4..4..1....0..2..2..3

R. H. Hardin, Table of n, a(n) for n = 1..180

Diagonal is A068748
Column 1 is A000244(n-1)
Column 2 is A138977
Column 3 is A138978
Column 4 is A138979

Empirical for column k:
k=1: a(n) = 3*a(n-1)
k=2: a(n) = 7*a(n-1) -4*a(n-2)
k=3: a(n) = 16*a(n-1) -31*a(n-2) +10*a(n-3)
k=4: [order 10]
k=5: [order 25]
k=6: [order 70]

cp's OEIS Frontend

A068744 Number of potential flows in 3 X 3 array with integer velocities in -n..n, i.e., number of 3 X 3 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

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A068747 Number of potential flows in 6 X 6 array with integer velocities in -n..n, i.e., number of 6 X 6 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

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A068745 Number of potential flows in 4 X 4 array with integer velocities in -n..n, i.e., number of 4 X 4 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

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A068746 Number of potential flows in 5 X 5 array with integer velocities in -n..n, i.e., number of 5 X 5 arrays with adjacent elements differing by no more than n, counting arrays differing by a constant only once.

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A222169 T(n,k)=Number of nXk 0..4 arrays with entries increasing mod 5 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

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