A068745 -id:A068745 - 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.

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.

A125530 Number of n X n arrays with entries in 1..3 in which adjacent entries differ by 1 or less (adjacent means in x or y directions).

Original entry on oeis.org

1, 3, 35, 2021, 536453, 661407975, 3780952160847, 100240477650720585, 12324646884990466825513, 7027467861520236770627093435, 18582989792598188051914777397933755

Offset: 0

R. H. Hardin, Dec 31 2006

nonn

[Empirical] a(base+1,n,diff)=a(base,n,diff)+F(n,diff) for base>=2.diff.(n-1) and F(n,diff)= (n=2, A063496(diff+1)) (n=3, A068744(diff+1)) (n=4, A068745(diff+1)) (n=5, A068746(diff+1)) (n=6, A068747(diff+1))

A125531 Number of n X n arrays with entries in 1..4 in which adjacent entries differ by 1 or less (adjacent means in x or y directions).

Original entry on oeis.org

1, 4, 54, 3682, 1209830, 1909172038, 14441437270426, 523308354987531254, 90815542491916256949262, 75466682574077698691682950866, 300268405527350796472180876929042538

Offset: 0

R. H. Hardin, Dec 31 2006

nonn

[Empirical] a(base+1,n,diff)=a(base,n,diff)+F(n,diff) for base>=2.diff.(n-1) and F(n,diff)= (n=2, A063496(diff+1)) (n=3, A068744(diff+1)) (n=4, A068745(diff+1)) (n=5, A068746(diff+1)) (n=6, A068747(diff+1))

A125532 Number of n X n arrays with entries in 1..4 in which adjacent entries differ by 2 or less (adjacent means in x or y directions).

Original entry on oeis.org

1, 4, 162, 75398, 386994338, 22039625597766, 13914188992963436802, 97391265572623379728784646, 7557580190940015054807296049439074, 6502004785230840000033839573162175747164038

Offset: 0

R. H. Hardin, Dec 31 2006

nonn

[Empirical] a(base+1,n,diff)=a(base,n,diff)+F(n,diff) for base>=2.diff.(n-1) and F(n,diff)= (n=2, A063496(diff+1)) (n=3, A068744(diff+1)) (n=4, A068745(diff+1)) (n=5, A068746(diff+1)) (n=6, A068747(diff+1))

A125533 Number of n X n arrays with entries in 1..5 in which adjacent entries differ by 2 or less (adjacent means in x or y directions).

Original entry on oeis.org

1, 5, 247, 156579, 1191861707, 109495018399901, 121266251495154433997, 1619036215215662119089701145, 260573722827793207854814217971985907

Offset: 0

R. H. Hardin, Dec 31 2006

nonn

[Empirical] a(base+1,n,diff)=a(base,n,diff)+F(n,diff) for base>=2.diff.(n-1) and F(n,diff)= (n=2, A063496(diff+1)) (n=3, A068744(diff+1)) (n=4, A068745(diff+1)) (n=5, A068746(diff+1)) (n=6, A068747(diff+1)).

A125534 Number of n X n arrays with entries in 1..5 in which adjacent entries differ by 3 or less (adjacent means in x or y directions).

Original entry on oeis.org

1, 5, 467, 877547, 32392165445, 23566800979324529, 337794127866926817026015, 95393532181581177183563991168527, 530759903230981312601669469700498881710153

Offset: 0

R. H. Hardin, Dec 31 2006

nonn

[Empirical] a(base+1,n,diff)=a(base,n,diff)+F(n,diff) for base>=2.diff.(n-1) and F(n,diff)= (n=2, A063496(diff+1)) (n=3, A068744(diff+1)) (n=4, A068745(diff+1)) (n=5, A068746(diff+1)) (n=6, A068747(diff+1))

A125535 Number of n X n arrays with entries in 1..6 in which adjacent entries differ by 3 or less (adjacent means in x or y directions).

Original entry on oeis.org

1, 6, 694, 1842302, 106627685414, 135334709216482502, 3763490062900640355493914, 2293235289704399729849626764238522, 30618002869205254395943653838381933205254046

Offset: 0

R. H. Hardin, Dec 31 2006

nonn

[Empirical] a(base+1,n,diff)=a(base,n,diff)+F(n,diff) for base>=2.diff.(n-1) and F(n,diff)= (n=2, A063496(diff+1)) (n=3, A068744(diff+1)) (n=4, A068745(diff+1)) (n=5, A068746(diff+1)) (n=6, A068747(diff+1))

A125536 Number of n X n arrays with entries in 1..6 in which adjacent entries differ by 4 or less (adjacent means in x or y directions).

Original entry on oeis.org

1, 6, 1058, 5748554, 948155870210, 4756811244059748042, 725704975038742796696062050, 3366853797682513115652233385161124618, 475014727091441236705100922925411926217680828034

Offset: 0

R. H. Hardin, Dec 31 2006

nonn

[Empirical] a(base+1,n,diff)=a(base,n,diff)+F(n,diff) for base>=2.diff.(n-1) and F(n,diff)= (n=2, A063496(diff+1)) (n=3, A068744(diff+1)) (n=4, A068745(diff+1)) (n=5, A068746(diff+1)) (n=6, A068747(diff+1))

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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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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A125530 Number of n X n arrays with entries in 1..3 in which adjacent entries differ by 1 or less (adjacent means in x or y directions).

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A125531 Number of n X n arrays with entries in 1..4 in which adjacent entries differ by 1 or less (adjacent means in x or y directions).

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A125532 Number of n X n arrays with entries in 1..4 in which adjacent entries differ by 2 or less (adjacent means in x or y directions).

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A125533 Number of n X n arrays with entries in 1..5 in which adjacent entries differ by 2 or less (adjacent means in x or y directions).

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A125534 Number of n X n arrays with entries in 1..5 in which adjacent entries differ by 3 or less (adjacent means in x or y directions).

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A125535 Number of n X n arrays with entries in 1..6 in which adjacent entries differ by 3 or less (adjacent means in x or y directions).

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A125536 Number of n X n arrays with entries in 1..6 in which adjacent entries differ by 4 or less (adjacent means in x or y directions).

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