A068747 -id:A068747 - 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

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.

A068761 Number of potential flows in n X n array with integer velocities in -14..14, i.e., number of n X n arrays with adjacent elements differing by no more than 14, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 16269, 105227044417, 278175212679968120043

Offset: 1

R. H. Hardin, Feb 27 2002

nonn

By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, solenoidal flows A068722-A068738.

A068748 Number of potential flows in n X n array with integer velocities in -1..1, i.e., number of n X n arrays with adjacent elements differing by no more than 1, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 19, 1665, 690437, 1366395515, 12961320464027, 590771311004409091, 129585582154154757804255, 136932044780329080616264070553, 697547669137459516115242710064880165

Offset: 1

R. H. Hardin, Feb 27 2002

nonn

By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, solenoidal flows A068722-A068738.

A068752 Number of potential flows in n X n array with integer velocities in -5..5, i.e., number of n X n arrays with adjacent elements differing by no more than 5, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 891, 45642289, 138190481342321, 24953087551680958151267

Offset: 1

R. H. Hardin, Feb 27 2002

nonn

By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, A068752 solenoidal flows A068722-A068738.

A068749 Number of potential flows in n X n array with integer velocities in -2..2, i.e., number of n X n arrays with adjacent elements differing by no more than 2, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 85, 87825, 1133641543, 184422574177355, 379827509377146575439, 9928923943603018785634645661

Offset: 1

R. H. Hardin, Feb 27 2002

nonn

By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, solenoidal flows A068722-A068738.

A068750 Number of potential flows in n X n array with integer velocities in -3..3, i.e., number of n X n arrays with adjacent elements differing by no more than 3, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 231, 1253329, 164185416899, 523957519578572209, 40918963843906494595507077

Offset: 1

R. H. Hardin, Feb 27 2002

nonn

By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, solenoidal flows A068722-A068738.

A068751 Number of potential flows in n X n array with integer velocities in -4..4, i.e., number of n X n arrays with adjacent elements differing by no more than 4, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 489, 9230193, 6913624013061, 207345516734034667209

Offset: 1

R. H. Hardin, Feb 27 2002

nonn

By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, solenoidal flows A068722-A068738.

A068753 Number of potential flows in n X n array with integer velocities in -6..6, i.e., number of n X n arrays with adjacent elements differing by no more than 6, counting arrays differing by a constant only once.

Original entry on oeis.org

1, 1469, 172989921, 1678843050246451, 1354915464537160758459123

Offset: 1

R. H. Hardin, Feb 27 2002

nonn

By size 3 X 3, ..., 6 X 6 A068744-A068747, by velocity limit 1..14 A068248-A068761, solenoidal flows A068722-A068738.

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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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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A068761 Number of potential flows in n X n array with integer velocities in -14..14, i.e., number of n X n arrays with adjacent elements differing by no more than 14, counting arrays differing by a constant only once.

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A068748 Number of potential flows in n X n array with integer velocities in -1..1, i.e., number of n X n arrays with adjacent elements differing by no more than 1, counting arrays differing by a constant only once.

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A068752 Number of potential flows in n X n array with integer velocities in -5..5, i.e., number of n X n arrays with adjacent elements differing by no more than 5, counting arrays differing by a constant only once.

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A068749 Number of potential flows in n X n array with integer velocities in -2..2, i.e., number of n X n arrays with adjacent elements differing by no more than 2, counting arrays differing by a constant only once.

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A068750 Number of potential flows in n X n array with integer velocities in -3..3, i.e., number of n X n arrays with adjacent elements differing by no more than 3, counting arrays differing by a constant only once.

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A068751 Number of potential flows in n X n array with integer velocities in -4..4, i.e., number of n X n arrays with adjacent elements differing by no more than 4, counting arrays differing by a constant only once.

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A068753 Number of potential flows in n X n array with integer velocities in -6..6, i.e., number of n X n arrays with adjacent elements differing by no more than 6, counting arrays differing by a constant only once.

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