A359573 -id:A359573 - cp's OEIS frontend

A359574 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with all 1's connected and a path of 1's from top row to bottom row.

1, 3, 1, 6, 7, 1, 10, 28, 17, 1, 15, 88, 144, 41, 1, 21, 245, 920, 730, 99, 1, 28, 639, 5191, 9362, 3692, 239, 1, 36, 1608, 27651, 104989, 94280, 18666, 577, 1, 45, 3968, 143342, 1111283, 2075271, 947760, 94384, 1393, 1, 55, 9689, 733512, 11457514, 42972329, 40792921, 9528128, 477264, 3363, 1

Offset: 1

Andrew Howroyd, Jan 06 2023

The grid has m rows and n columns.

			Array begins:
================================================================
m\n| 1   2     3       4         5           6             7
---+------------------------------------------------------------
1  | 1   3     6      10        15          21            28 ...
2  | 1   7    28      88       245         639          1608 ...
3  | 1  17   144     920      5191       27651        143342 ...
4  | 1  41   730    9362    104989     1111283      11457514 ...
5  | 1  99  3692   94280   2075271    42972329     866126030 ...
6  | 1 239 18666  947760  40792921  1642690309   64270256276 ...
7  | 1 577 94384 9528128 801218515 62618577481 4741764527414 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..435
Chaim Goodman-Strauss, Notes on the number of m × n binary arrays with all 1's connected and a path of 1's from top row to bottom row (May 21 2020)

Main diagonal is A163028.
Rows 1..10 are A000217, A163037, A163038, A163039, A163040, A163041, A163042, A163043, A163044, A163045.
Columns 1..10 are A000012, A001333(n+1), A163029, A163030, A163031, A163032, A163033, A163034, A163035, A163036.
Cf. A287151, A292357, A359573, A359576.

T(m,n) = A287151(m,n) - 2*A287151(m-1,n) + A287151(m-2,n) for m > 2.

A359575 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with a path of adjacent 1's from upper right corner to lower left corner.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 20, 51, 20, 1, 1, 49, 295, 295, 49, 1, 1, 119, 1632, 3828, 1632, 119, 1, 1, 288, 8830, 46557, 46557, 8830, 288, 1, 1, 696, 47239, 546286, 1225194, 546286, 47239, 696, 1, 1, 1681, 251261, 6279393, 30754544, 30754544, 6279393, 251261, 1681, 1

Offset: 1

Andrew Howroyd, Jan 06 2023

			Array begins:
================================================================
m\n| 1   2     3       4         5           6             7
---+------------------------------------------------------------
1  | 1   1     1       1         1           1             1 ...
2  | 1   3     8      20        49         119           288 ...
3  | 1   8    51     295      1632        8830         47239 ...
4  | 1  20   295    3828     46557      546286       6279393 ...
5  | 1  49  1632   46557   1225194    30754544     749866185 ...
6  | 1 119  8830  546286  30754544  1636193228   83949041929 ...
7  | 1 288 47239 6279393 749866185 83949041929 9009490924794 ...
  ...

Main diagonal is A069343.
Rows 1..20 are A000012, A048739(n-1), A069325, A069326, A069327-A069342.
Cf. A359573, A359576.

T(m,n) = T(n,m).

A163002 Number of n X n binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

Original entry on oeis.org

1, 3, 45, 2423, 482443, 360923899, 1012326365581, 10583931453268587, 410628432970492896401, 58940807698860767759080097, 31235684611422395105793077154373, 61025622782775877079925551890952450707, 439058030866550268551946597005515209497083151

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..15

Main diagonal of A359573.

a(8)-a(11) from Andrew Howroyd, Jan 06 2023

Terms a(12) and beyond from Andrew Howroyd, Jan 30 2023

A163003 Number of n X 3 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

Original entry on oeis.org

1, 8, 45, 234, 1193, 6049, 30616, 154861, 783150, 3960215, 20025500, 101261654, 512042141, 2589202874, 13092613725, 66204360820, 334770228305, 1692805481383, 8559872259604, 43284012160451, 218870755473698, 1106746006355201, 5596392811360844, 28298825854292256

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..100

Row 3 of A359573.

From R. J. Mathar, Aug 11 2009: (Start)
Empirical: a(n) = 8*a(n-1) -17*a(n-2) +10*a(n-3) +6*a(n-4) -14*a(n-5) +7*a(n-6) +6*a(n-7) -a(n-8).
Empirical g.f.: -x*(1+x)*(x^4-x^3+x^2+x-1)/ ((x^2+x-1)*(x^6-7*x^5+x^4+6*x^3-11*x^2+7*x-1)). (End)
The above empirical formulas are correct. - Andrew Howroyd, Jan 06 2023

A163004 Number of n X 4 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

Original entry on oeis.org

1, 20, 234, 2423, 24455, 245972, 2473317, 24868730, 250043151, 2514035445, 25277060955, 254144975454, 2555267942068, 25691612225457, 258313003017344, 2597174783771339, 26112958991681036, 262549379235881855

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..44

Row 4 of A359573.

A163005 Number of n X 5 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

Original entry on oeis.org

1, 49, 1193, 24455, 482443, 9469361, 185899132, 3650755085, 71703384184, 1408344201283, 27661780849317, 543315030292163, 10671446719498293, 209601732562410408, 4116863153169917525, 80860792337380041052

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

R. H. Hardin, Table of n, a(n) for n=1..17

Row 5 of A359573.

A163006 Number of n X 6 binary arrays with all 1's connected and a path of 1's from upper left corner to lower right corner.

Original entry on oeis.org

1, 119, 6049, 245972, 9469361, 360923899, 13742823032, 523375860237, 19934900617863, 759343825445905, 28924772690242607, 1101801030283339240, 41969783540217156334, 1598712332667377324778, 60898124432179640663437, 2319730350681751316658034, 88363129835202482890093852

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row 6 of A359573.

Terms a(13) and beyond from Andrew Howroyd, Jan 06 2023

A163007 Number of n X 7 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

Original entry on oeis.org

1, 288, 30616, 2473317, 185899132, 13742823032, 1012326365581, 74513820639189, 5483824484276487, 403566354707763962, 29699049493529506304, 2185591507222956126904, 160840358014084186505186, 11836434997659016450700108, 871057350099281882543862533

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row 7 of A359573.

Terms a(10) and beyond from Andrew Howroyd, Jan 06 2023

A163008 Number of n X 8 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

Original entry on oeis.org

1, 696, 154861, 24868730, 3650755085, 523375860237, 74513820639189, 10583931453268587, 1502037917021311844, 213092418506218124956, 30227108359825419597288, 4287476337332745216533644, 608131459043610000423372550, 86256029502930854834545052267

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row 8 of A359573.

Terms a(8) and beyond from Andrew Howroyd, Jan 06 2023

A163009 Number of n X 9 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

Original entry on oeis.org

1, 1681, 783150, 250043151, 71703384184, 19934900617863, 5483824484276487, 1502037917021311844, 410628432970492896401, 112160245091369049581311, 30623519350300448308681813, 8359714215859160247276696744, 2281871672070839340863159155338, 622836891043514813722341456961744

Offset: 1

R. H. Hardin, Jul 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row 9 of A359573.

Terms a(8) and beyond from Andrew Howroyd, Jan 06 2023

cp's OEIS Frontend

A359574 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with all 1's connected and a path of 1's from top row to bottom row.

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A359575 Array read by antidiagonals: T(m,n) is the number of m X n binary arrays with a path of adjacent 1's from upper right corner to lower left corner.

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A163002 Number of n X n binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

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A163003 Number of n X 3 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

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A163004 Number of n X 4 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

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A163005 Number of n X 5 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

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A163006 Number of n X 6 binary arrays with all 1's connected and a path of 1's from upper left corner to lower right corner.

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A163007 Number of n X 7 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

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A163008 Number of n X 8 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

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A163009 Number of n X 9 binary arrays with all 1s connected and a path of 1s from upper left corner to lower right corner.

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