cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A068722 Number of solenoidal flows (flow in = flow out) in a 3 X 3 square array with integer velocities -n .. n.

Original entry on oeis.org

1, 35, 247, 925, 2501, 5551, 10795, 19097, 31465, 49051, 73151, 105205, 146797, 199655, 265651, 346801, 445265, 563347, 703495, 868301, 1060501, 1282975, 1538747, 1830985, 2163001, 2538251, 2960335, 3432997, 3960125, 4545751, 5194051, 5909345, 6696097, 7558915
Offset: 0

Views

Author

R. H. Hardin, Feb 26 2002

Keywords

Comments

Conjecture: A255437(a(n)) = 2*n+1, i.e. a(n) = gives the position of the first occurrence of 2*n+1 in A255437. - Reinhard Zumkeller, Mar 23 2015

Examples

			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.
3 X 3 cell centers are at the intersection of long rows and long columns.
n=1:
.. 0 . 0 . 0
.0. -1. -1 . 0
.. 1 . 0. -1
.0 . 0 . 0 . 0
.. 1 . 0. -1
.0 . 1 . 1 . 0
.. 0 . 0 . 0
n=2:
.. 0 . 0 . 0
.0. -2. -1 . 0
.. 2. -1. -1
.0 . 0. -1 . 0
.. 2 . 0. -2
.0 . 2 . 2 . 0
.. 0 . 0 . 0

Links

Crossrefs

Cf. 2 X 2=1, 3, 5, 7..., 4 X 4 A068723, 5 X 5 A068724, 6x6 A068725, by velocity limit 1..13 A068726-A068738.
Cf. A016754, A255437.

Programs

Formula

a(n) = (1+2*n+2*n^2) * (1+3*n+3*n^2).
G.f.: (1+30*x+82*x^2+30*x^3+x^4)/(1-x)^5. - Colin Barker, Jul 30 2012
E.g.f.: exp(x)*(1 + 34*x + 89*x^2 + 48*x^3 + 6*x^4). - Stefano Spezia, Mar 10 2024

Extensions

Formula corrected by Colin Barker, Jul 30 2012

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

Views

Author

R. H. Hardin, Feb 26 2002

Keywords

Examples

			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

Crossrefs

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.

Extensions

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

A068723 Number of solenoidal flows (flow in = flow out) in a 4 X 4 square array with integer velocities -n .. n.

Original entry on oeis.org

1, 2053, 160403, 3096791, 28905785, 173429597, 773568107, 2789847759, 8576204625, 23279942261, 57201104003, 129540052071, 274074872265, 547436323149, 1040787422427, 1895867410719, 3326525754145, 5647051042021, 9308792104563, 14946774417783, 23438233878745, 35975222324029, 54153684727627, 80081666851567, 116509584233329
Offset: 0

Views

Author

R. H. Hardin, Feb 26 2002

Keywords

Examples

			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.
4 X 4 cell centers are at the intersection of long rows and long columns.
n=1:
.. 0 . 0 . 0 . 0
.0. -1. -1. -1 . 0
.. 1 . 0 . 0. -1
.0 . 0. -1 . 0 . 0
.. 1 . 1. -1. -1
.0 . 0 . 1 . 0 . 0
.. 1 . 0 . 0. -1
.0 . 1 . 1 . 1 . 0
.. 0 . 0 . 0 . 0
n=2:
.. 0 . 0 . 0 . 0
.0. -2. -1. -2 . 0
.. 2. -1 . 1. -2
.0 . 0. -2 . 0 . 0
.. 2 . 1. -1. -2
.0 . 0 . 1 . 0 . 0
.. 2 . 0 . 0. -2
.0 . 2 . 2 . 2 . 0
.. 0 . 0 . 0 . 0

Links

Crossrefs

Cf. A005408 (2 X 2), A068722 (3 X 3), A068724 (5 X 5), A068725 (6 X 6).
By velocity limit 1..13 A068726-A068738.

Formula

Empirical G.f.: (x+1) * (x^8 + 2042*x^7 + 137876*x^6 + 1447150*x^5 + 3462710*x^4 + 1447150*x^3 + 137876*x^2 + 2042*x + 1) / (x-1)^10. [Colin Barker, Jul 31 2012]

Extensions

a(14)-a(24) from Sean A. Irvine, Mar 06 2024
Showing 1-3 of 3 results.

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