A068722 -id:A068722 - cp's OEIS frontend

A068738 Number of solenoidal flows (flow in = flow out) in an n X n square array with integer velocities in -13 .. 13.

Original entry on oeis.org

1, 27, 199655, 547436323149, 535691029084640060219

Offset: 1

R. H. Hardin, Feb 26 2002

By size 3 X 3, ..., 6 X 6 A068722-A068725, by velocity limit 1..13 A068726-A068738.

a(5) from Sean A. Irvine, Mar 10 2024

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.

Original entry on oeis.org

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.

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.

A068726 Number of solenoidal flows (flow in = flow out) in an n X n square array with integer velocities in -1 .. 1.

Original entry on oeis.org

1, 3, 35, 2053, 569397, 758541101, 4855339973053, 149596683621380451, 22207207625437795523567, 15894065026795563049567474623, 54873022990102933765115144179983347, 914167438412564688754864863927964270928677, 73512018227635160780006728165405332017240587741389

Offset: 1

R. H. Hardin, Feb 26 2002

nonn

By size 3 X 3, ..., 6 X 6 A068722-A068725, by velocity limit 1..13 A068726-A068738.

a(12)-a(13) from Sean A. Irvine, Mar 06 2024

A068725 Number of solenoidal flows (flow in = flow out) in a 6x6 square array with integer velocities -n .. n.

Original entry on oeis.org

1, 758541101, 134771061592765, 500885416261650533, 247753867499441888279, 35922392489639958547367, 2286173927365217148894607, 80633423581611695455020595

Offset: 0

R. H. Hardin, Feb 26 2002

			Sample flows (. represents a space):
Numbers in long rows are on cell walls showing velocity rightward.
Numbers in long columns are on cell floors showing velocity downwards.
6x6 cell centers are at the intersection of long rows and long columns.
n=1:
.. 0 . 0 . 0 . 0 . 0 . 0
.0. -1. -1. -1. -1. -1 . 0
.. 1 . 0 . 0 . 0 . 0. -1
.0 . 0. -1. -1. -1 . 0 . 0
.. 1 . 1 . 0 . 0. -1. -1
.0 . 0 . 0. -1 . 0 . 0 . 0
.. 1 . 1 . 1. -1. -1. -1
.0 . 0 . 0 . 1 . 0 . 0 . 0
.. 1 . 1 . 0 . 0. -1. -1
.0 . 0 . 1 . 1 . 1 . 0 . 0
.. 1 . 0 . 0 . 0 . 0. -1
.0 . 1 . 1 . 1 . 1 . 1 . 0
.. 0 . 0 . 0 . 0 . 0 . 0
n=2:
.. 0 . 0 . 0 . 0 . 0 . 0
.0. -2. -1. -1 . 0. -1 . 0
.. 2. -1 . 0. -1 . 1. -1
.0 . 0. -2. -2. -2. -1 . 0
.. 2 . 1 . 0. -1 . 0. -2
.0 . 0. -1. -2. -2 . 0 . 0
.. 2 . 2 . 1. -1. -2. -2
.0 . 0 . 0 . 1 . 0 . 0 . 0
.. 2 . 2 . 0 . 0. -2. -2
.0 . 0 . 2 . 2 . 2 . 0 . 0
.. 2 . 0 . 0 . 0 . 0. -2
.0 . 2 . 2 . 2 . 2 . 2 . 0
.. 0 . 0 . 0 . 0 . 0 . 0

2 X 2 gives 1, 3, 5, 7..., 3 X 3 gives A068722, 4 X 4 gives A068723, 5 X 5 gives A068724, by velocity limit 1..13 A068726-A068738.

a(4)-a(7) from Sean A. Irvine, Mar 07 2024

A255437 In positive integers: replace k^2 with the first k odd numbers.

Original entry on oeis.org

1, 2, 3, 1, 3, 5, 6, 7, 8, 1, 3, 5, 10, 11, 12, 13, 14, 15, 1, 3, 5, 7, 17, 18, 19, 20, 21, 22, 23, 24, 1, 3, 5, 7, 9, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 1, 3, 5, 7, 9, 11, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 1, 3, 5, 7, 9, 11, 13, 50, 51

Offset: 1

Reinhard Zumkeller, Mar 23 2015

nonn

a(A005448(n)) = 1;
conjecture: a(A068722(n)) = (2*n+1)^2, i.e. A068722(n) = gives the position of the first occurrence of n-th odd square;
A164514(n) = a(A255527(n)) and a(m) < A164514(n) for m < A255527(n).

			.  A000290 | 1,    4,          9,                      16,         . . .
.  A000027 | _,2,3,___,5,6,7,8,_____,10,11,12,13,14,15,_______,17,18,...
.  A158405 | 1,    1,3,        1,3,5,                  1,3,5,7,
.  --------+-------------------------------------------------------------
.     a(n) | 1,2,3,1,3,5,6,7,8,1,3,5,10,11,12,13,14,15,1,3,5,7,17,18,19 .

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A256188, A000290, A000037, A158405, A016742, A164514, A255527, A005448, A255507 (first differences), A255508 (partial sums).

a255437 n = a255437_list !! (n-1)
a255437_list = f 0 [1..] a158405_tabl where
   f k xs (zs:zss) = us ++ zs ++ f (k + 2) vs zss
                     where (us, v:vs) = splitAt k xs

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.

cp's OEIS Frontend

A068738 Number of solenoidal flows (flow in = flow out) in an n X n square array with integer velocities in -13 .. 13.

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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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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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A068726 Number of solenoidal flows (flow in = flow out) in an n X n square array with integer velocities in -1 .. 1.

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A068725 Number of solenoidal flows (flow in = flow out) in a 6x6 square array with integer velocities -n .. n.

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A255437 In positive integers: replace k^2 with the first k odd numbers.

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Haskell

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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