A229428 -id:A229428 - cp's OEIS frontend

A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

0, 0, 0, 1, 0, 1, 3, 4, 4, 3, 6, 16, 25, 16, 6, 10, 41, 94, 94, 41, 10, 15, 85, 266, 386, 266, 85, 15, 21, 155, 632, 1247, 1247, 632, 155, 21, 28, 259, 1332, 3423, 4657, 3423, 1332, 259, 28, 36, 406, 2570, 8342, 14795, 14795, 8342, 2570, 406, 36, 45, 606, 4631, 18546, 41586, 54219, 41586, 18546, 4631, 606, 45

Offset: 1

N. J. A. Sloane, Feb 06 2019

The monotonicity condition requires that M_{(i+1),j} = M_{i,j} + (0 or 1); M_{i,(j+1)} = M_{i,j} + (0 or 1).
These matrices can be cut into three connected pieces, containing the 0's, 1's, and 2's; there are two vertex-disjoint paths from the north-and-east edges of the matrix to the south-and-west edges.
Row (or column) n >= 1 has a linear recurrence (with constant coefficients) of order 2n+1. - Alois P. Heinz, Feb 07 2019

			Array begins:
    0   0    1    3     6    10 ...
    0   0    4   16    41    85 ...
    1   4   25   94   266   632 ...
    3  16   94  386  1247  3423 ...
    6  41  266 1247  4657 14795 ...
   10  85  632 3427 14795 54219 ...
...
The 4 examples when m=2 and n=3 are
    011   011  012   012
    012   112  012   112

D. E. Knuth, Email to N. J. A. Sloane, Feb 05 2019.

Alois P. Heinz, Antidiagonals n = 1..80, flattened

Rows 1-10 give: A000217(n-2), A323847, A323967, A323968, A323969, A323970, A323971, A323972, A323973, A323974.
Main diagonal gives A306322.
Cf. A132823, A252876, A229428.

More terms from Alois P. Heinz, Feb 07 2019

A229421 Number of n X n 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

3, 12, 83, 754, 7816, 87648, 1037193, 12768612, 162034856, 2106287556, 27919532994, 376116911288, 5136301331502, 70961712401226, 990271481453131, 13940677426765198, 197768065242681646, 2824828320413979786, 40595088421065998006, 586585037559246948950

Offset: 1

R. H. Hardin, Sep 22 2013

nonn

			Some solutions for n=4
..1..1..0..0....1..1..1..1....0..0..0..0....2..2..1..1....2..1..1..0
..1..1..0..0....2..1..1..1....1..1..0..0....2..2..2..1....2..1..1..0
..2..1..1..0....2..2..1..1....2..1..1..1....2..2..2..1....2..1..1..1
..2..1..1..0....2..2..2..1....2..2..2..1....2..2..2..2....2..2..2..2

Alois P. Heinz, Table of n, a(n) for n = 1..40

Diagonal of A229428.

a(16)-a(20) from Alois P. Heinz, Feb 08 2019

A229422 Number of n X 2 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

5, 12, 27, 55, 102, 175, 282, 432, 635, 902, 1245, 1677, 2212, 2865, 3652, 4590, 5697, 6992, 8495, 10227, 12210, 14467, 17022, 19900, 23127, 26730, 30737, 35177, 40080, 45477, 51400, 57882, 64957, 72660, 81027, 90095, 99902, 110487, 121890, 134152

Offset: 1

R. H. Hardin, Sep 22 2013

nonn

			Some solutions for n=4:
..0..0....0..0....1..0....0..0....1..1....1..0....0..0....0..0....2..2....0..0
..0..0....1..0....1..0....0..0....2..1....2..1....0..0....1..1....2..2....1..1
..1..1....1..1....1..1....0..0....2..1....2..1....1..0....1..1....2..2....2..2
..2..2....2..1....2..1....1..1....2..2....2..1....1..0....2..1....2..2....2..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 2 of A229428.

Empirical: a(n) = (1/24)*n^4 + (5/12)*n^3 + (11/24)*n^2 + (25/12)*n + 2.
Conjectures from Colin Barker, Sep 15 2018: (Start)
G.f.: x*(5 - 13*x + 17*x^2 - 10*x^3 + 2*x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A229423 Number of n X 3 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

8, 27, 83, 222, 524, 1116, 2187, 4005, 6936, 11465, 18219, 27992, 41772, 60770, 86451, 120567, 165192, 222759, 296099, 388482, 503660, 645912, 820091, 1031673, 1286808, 1592373, 1956027, 2386268, 2892492, 3485054, 4175331, 4975787, 5900040, 6962931

Offset: 1

R. H. Hardin, Sep 22 2013

nonn

			Some solutions for n=4:
..1..0..0....1..1..0....0..0..0....1..1..1....1..0..0....0..0..0....1..0..0
..2..1..1....2..1..0....1..1..0....1..1..1....2..1..1....1..1..0....2..1..0
..2..1..1....2..2..1....1..1..1....1..1..1....2..2..1....1..1..0....2..1..1
..2..2..1....2..2..1....1..1..1....1..1..1....2..2..2....1..1..1....2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 3 of A229428.

Empirical: a(n) = (1/360)*n^6 + (1/20)*n^5 + (5/18)*n^4 + (2/3)*n^3 + (799/360)*n^2 + (107/60)*n + 3.
Conjectures from Colin Barker, Sep 15 2018: (Start)
G.f.: x*(8 - 29*x + 62*x^2 - 72*x^3 + 48*x^4 - 18*x^5 + 3*x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A229424 Number of n X 4 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

12, 55, 222, 754, 2204, 5700, 13345, 28794, 58053, 110550, 200533, 348855, 585211, 950897, 1502166, 2314261, 3486210, 5146473, 7459536, 10633552, 14929134, 20669410, 28251455, 38159220, 50978083, 67411152, 88297455, 114632157

Offset: 1

R. H. Hardin, Sep 22 2013

nonn

			Some solutions for n=4:
..2..2..1..0....1..1..0..0....2..1..0..0....1..1..1..0....1..0..0..0
..2..2..2..1....2..1..0..0....2..2..1..1....2..1..1..0....2..1..0..0
..2..2..2..1....2..1..1..1....2..2..1..1....2..2..1..0....2..1..1..1
..2..2..2..1....2..2..2..2....2..2..2..2....2..2..2..1....2..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 4 of A229428.

Empirical: a(n) = (1/8064)*n^8 + (1/288)*n^7 + (103/2880)*n^6 + (17/90)*n^5 + (751/1152)*n^4 + (571/288)*n^3 + (7769/3360)*n^2 + (153/40)*n + 3.
Conjectures from Colin Barker, Sep 15 2018: (Start)
G.f.: x*(12 - 53*x + 159*x^2 - 272*x^3 + 302*x^4 - 222*x^5 + 103*x^6 - 27*x^7 + 3*x^8) / (1 - x)^9.
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9) for n>9.
(End)

A229425 Number of n X 5 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

17, 102, 524, 2204, 7816, 24126, 66503, 166972, 387738, 842802, 1731129, 3385828, 6344979, 11451106, 19987862, 33864276, 55858927, 89938670, 141669058, 218736396, 331604442, 494332150, 725582545, 1049856844, 1498992304

Offset: 1

R. H. Hardin, Sep 22 2013

nonn

			Some solutions for n=4:
..0..0..0..0..0....1..0..0..0..0....1..1..0..0..0....2..2..1..1..0
..0..0..0..0..0....1..1..0..0..0....1..1..0..0..0....2..2..1..1..0
..1..1..1..1..0....2..1..1..0..0....2..1..1..1..0....2..2..2..2..1
..2..1..1..1..0....2..2..1..0..0....2..2..1..1..0....2..2..2..2..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Column 5 of A229428.

Empirical: a(n) = (1/259200)*n^10 + (1/6480)*n^9 + (149/60480)*n^8 + (163/7560)*n^7 + (10411/86400)*n^6 + (1063/2160)*n^5 + (19597/12960)*n^4 + (4169/1620)*n^3 + (30649/6300)*n^2 + (215/63)*n + 4.
Conjectures from Colin Barker, Sep 15 2018: (Start)
G.f.: x*(17 - 85*x + 337*x^2 - 755*x^3 + 1172*x^4 - 1284*x^5 + 987*x^6 - 525*x^7 + 186*x^8 - 40*x^9 + 4*x^10) / (1 - x)^11.
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>11.
(End)

A229426 Number of n X 6 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

23, 175, 1116, 5700, 24126, 87648, 281016, 812352, 2152643, 5297329, 12231874, 26724490, 55625600, 110928984, 212948248, 395089316, 710860767, 1243965435, 2122565928, 3539121568, 5777563514, 9250018192, 14545817232, 22496156480

Offset: 1

R. H. Hardin, Sep 22 2013

nonn

Column 6 of A229428.

			Some solutions for n=4
..2..2..1..0..0..0....1..0..0..0..0..0....2..2..1..1..1..0....2..2..2..1..1..1
..2..2..1..1..1..0....2..1..0..0..0..0....2..2..1..1..1..0....2..2..2..1..1..1
..2..2..1..1..1..0....2..1..0..0..0..0....2..2..2..1..1..0....2..2..2..1..1..1
..2..2..2..1..1..0....2..2..1..1..0..0....2..2..2..1..1..0....2..2..2..2..1..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A229428.

Empirical: a(n) = (1/11404800)*n^12 + (1/211200)*n^11 + (37/345600)*n^10 + (143/103680)*n^9 + (27977/2419200)*n^8 + (83399/1209600)*n^7 + (321133/1036800)*n^6 + (35291/34560)*n^5 + (66941/28800)*n^4 + (614843/129600)*n^3 + (149511/30800)*n^2 + (19627/3465)*n + 4.

A229427 Number of nX7 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

Original entry on oeis.org

30, 282, 2187, 13345, 66503, 281016, 1037193, 3420692, 10260128, 28379127, 73192023, 177601008, 408454945, 895812869, 1883191437, 3811219431, 7453126082, 14128822736, 26035706883, 46749593643, 81969420795, 140605855270

Offset: 1

R. H. Hardin Sep 22 2013

nonn

Column 7 of A229428

			Some solutions for n=4
..1..1..0..0..0..0..0....1..1..1..1..0..0..0....1..1..0..0..0..0..0
..1..1..0..0..0..0..0....2..1..1..1..0..0..0....1..1..1..1..0..0..0
..1..1..0..0..0..0..0....2..2..2..1..1..0..0....2..2..1..1..1..1..1
..1..1..0..0..0..0..0....2..2..2..1..1..0..0....2..2..2..2..2..2..1

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = (1/660441600)*n^14 + (1/9434880)*n^13 + (1/311850)*n^12 + (283/4989600)*n^11 + (23/34560)*n^10 + (5/896)*n^9 + (55843/1587600)*n^8 + (154153/907200)*n^7 + (4503127/7257600)*n^6 + (1244953/725760)*n^5 + (56059/14400)*n^4 + (1646347/302400)*n^3 + (401116003/50450400)*n^2 + (124079/24024)*n + 5

cp's OEIS Frontend

A323846 Array read by antidiagonals: T(m,n) = number of m X n matrices M with entries {0,1,2} that have M_{1,1}=0, M_{m,n}=2, are such that the rows and columns are monotonic without jumps of 2, and satisfy M_{(i+1),(j+1)} = M_{i,j} + (0 or 1).

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A229421 Number of n X n 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

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A229422 Number of n X 2 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

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A229423 Number of n X 3 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

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A229424 Number of n X 4 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

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A229425 Number of n X 5 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

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A229426 Number of n X 6 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

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A229427 Number of nX7 0..2 arrays with horizontal differences mod 3 never 1, vertical differences mod 3 never -1, rows lexicographically nondecreasing, and columns lexicographically nonincreasing.

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