A195220 -id:A195220 - cp's OEIS frontend

A195213 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 1 or less and triangles differing by a constant counted only once.

1, 7, 91, 2277, 111031, 10654607, 2021888119, 761408842173, 570411420721871, 851650667866314005, 2537743716708476573853, 15108768742290768272235707, 179885152873727967922701202921, 4286156683596201247493358770260469

Offset: 1

R. H. Hardin Sep 13 2011

nonn

Column 1 of A195220

			Some solutions for n=5
...0...............0...............0...............0...............0
...1..0...........-1.-1...........-1..0...........-1..0............1..1
...0..0..1........-2.-2.-1........-1..0..1.........0..0.-1.........0..1..0
...0..1..1..1.....-1.-1.-1.-2......0..0..0..1......1..0.-1.-2......0..1..1..0
...1..0..1..0..0..-1.-1.-2.-1.-2...0..0..1..1..2...1..0.-1.-2.-1...0..0..1..1..0

R. H. Hardin, Table of n, a(n) for n = 1..16

A195214 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 2 or less and triangles differing by a constant counted only once.

Original entry on oeis.org

1, 19, 1047, 176471, 92031109, 149824887097, 764465228592699, 12257873230432791803, 618761146282316127586145, 98449515559059678433241253935, 49416410864494101822050533251357921

Offset: 1

R. H. Hardin Sep 13 2011

nonn

Column 2 of A195220

			Some solutions for n=5
...0...............0...............0...............0...............0
...2..1...........-2.-1............1..0............0..1............0..1
...0..2..2.........0..0.-1.........0..2..2........-1..1..3.........1..0..0
...2..1..2..1......2..1..1..1......1..1..1..0......1..1..2..1.....-1.-1..1..2
...2..0..1..2..0...1..0..1..3..1...3..3..1..0..1...2..0..2..0..2..-2..0..0..2..2

R. H. Hardin, Table of n, a(n) for n = 1..12

A195221 Number of lower triangles of a 3 X 3 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

Original entry on oeis.org

91, 1047, 5453, 18903, 51205, 117585, 239891, 447797, 780007, 1285459, 2024529, 3070235, 4509441, 6444061, 8992263, 12289673, 16490579, 21769135, 28320565, 36362367, 46135517, 57905673, 71964379, 88630269, 108250271, 131200811

Offset: 1

R. H. Hardin, Sep 13 2011

nonn

Row 3 of A195220.

			Some solutions for n=5:
  0         0         0         0         0         0         0         0
  3  0      4  4      2  1     -5 -3     -4 -2     -4 -1     -1 -5      3  3
  0 -1 -2  -1 -1  4  -3  1 -2  -5 -4  0  -6 -6 -5   1  0 -4  -2 -5-10   8  4  8

R. H. Hardin, Table of n, a(n) for n = 1..200

Cf. A195220.

Empirical: a(n) = (301/30)*n^5 + (301/12)*n^4 + (88/3)*n^3 + (227/12)*n^2 + (199/30)*n + 1.
Empirical g.f.: x*(91 + 501*x + 536*x^2 + 70*x^3 + 7*x^4 - x^5) / (1 - x)^6. - Colin Barker, May 06 2018
Since a(n) is an Ehrhart polynomial of degree 5 (see A195220), and the empirical polynomial fits the Data for 1 <= n <= 6, it must be correct. - Robert Israel, Oct 06 2019

A195222 Number of lower triangles of a 4 X 4 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

Original entry on oeis.org

2277, 176471, 3395245, 31640829, 189677411, 845613769, 3048698613, 9369900047, 25430520379, 62478288201, 141479051231, 299315365855, 597820942629, 1136532061199, 2070203713171, 3632319367413, 6166018879097, 10164079181491

Offset: 1

R. H. Hardin, Sep 13 2011

nonn

Row 4 of A195220.

			Some solutions for n=5:
   0            0            0            0            0            0
  -5 -2         4  5         0  3         5  2        -4 -4         1 -4
  -2 -6 -1      0  3  2      5  0  5      3  0 -3     -5 -2 -1     -3  1 -2
  -3 -1 -6 -1   3  2  5  7   4  4  2  4   3  2  0  1  -7 -2 -6 -4   0 -2 -3 -6

R. H. Hardin, Table of n, a(n) for n = 1..194

Cf. A195220.

Empirical: a(n) = (1207573/30240)*n^9 + (1207573/6720)*n^8 + (1000157/2520)*n^7 + (264247/480)*n^6 + (754417/1440)*n^5 + (338651/960)*n^4 + (2533393/15120)*n^3 + (90763/1680)*n^2 + (901/84)*n + 1.

Since a(n) is an Ehrhart polynomial of degree 9 (see A195220), and the empirical polynomial fits the Data for 1 <= n <= 10, it must be correct. - Robert Israel, Oct 06 2019

A195223 Number of lower triangles of a 5 X 5 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

Original entry on oeis.org

111031, 92031109, 9032683465, 289301569283, 4677360495205, 47764170577925, 350767341744137, 2010235691940497, 9496465116615081, 38429133040711965, 136997589911672127, 439401533118090493, 1288688520518224397

Offset: 1

R. H. Hardin, Sep 13 2011

nonn

Row 5 of A195220.

			Some solutions for n=5:
  0               0               0               0               0
  2  4            5  2            3  3            1  5            1  4
  7  2  3         1  5  6         5  7  6         2  3  2        -1  0 -1
  5  4  3  7      2  4  1  1      3  8  7  3      5  6  3  0      0  3  0  2
  6  5  6  2  5   5  3  6  3  1   5  3  5  6  2   9  7  2  0  0   5  3 -1  4  3

R. H. Hardin, Table of n, a(n) for n = 1..43

Cf. A195220.

Empirical: a(n) = (3508493543/18345600)*n^14 + (3508493543/2620800)*n^13 + (1116775769537/239500800)*n^12 + (422094048023/39916800)*n^11 + (377328209183/21772800)*n^10 + (78475421219/3628800)*n^9 + (1073748492569/50803200)*n^8 + (19848770813/1209600)*n^7 + (221251862417/21772800)*n^6 + (18121075223/3628800)*n^5 + (10435002133/5443200)*n^4 + (505904317/907200)*n^3 + (8793472607/75675600)*n^2 + (1397863/90090)*n + 1.

Since a(n) is an Ehrhart polynomial of degree 14 (see A195220), and the empirical polynomial fits the Data for 1 <= n <= 15, it must be correct. - Robert Israel, Oct 06 2019

A195224 Number of lower triangles of a 6 X 6 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

Original entry on oeis.org

10654607, 149824887097, 103565705397639, 14572563308953245, 774355028021195459, 21369796486647028143, 368448539594713382337, 4459232627488479738855, 40959753617162679435381, 301639473232633417395629

Offset: 1

R. H. Hardin, Sep 13 2011

nonn

Row 6 of A195220.

a(n) is an Ehrhart polynomial of degree 20. - Robert Israel, Oct 07 2019

			Some solutions for n=5:
   0                  0                  0                  0
   1  1              -5 -3              -3 -4               3  2
   2  1  2           -1 -1 -5           -1 -1 -2            6  5  1
   4  0  3  0        -1 -2 -1 -5         4  2  0 -2         6  1  2 -2
   3 -1  4  4  1      0  1 -1 -3 -4     -1  4  2 -2 -5      1  2  1  0 -4
   0 -1  4  1  5  5   1 -4  1  0  1 -2  -1  2  1  0 -2 -5  -2  0  3 -1 -5 -3

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

A195225 Number of lower triangles of a 7X7 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

Original entry on oeis.org

2021888119, 764465228592699, 5137554141710397313, 4059527861103273129317, 863969905606597340602967, 76019997788404353762724447, 3546881146489163009435488735, 102667264655001518661047239817

Offset: 1

R. H. Hardin, Sep 13 2011

nonn

Row 7 of A195220

a(n) is an Ehrhart polynomial of degree 27. - Robert Israel, Oct 07 2019

			Some solutions for n=5
...0.....................0.....................0
...3..5..................5..0.................-3.-3
...7..8..9...............5..3..2..............-2..0.-1
...5..7..4..4............8..4.-1.-2...........-4.-4.-2..1
...6..4..4.-1.-1.........3..3.-1..0.-4........-6.-1.-1.-3..1
...1..4..2.-1..4..2......3..1.-2.-2.-3.-3.....-4.-4.-2.-4.-2..3
...6..3..4..0..4..6..3...4..3..0..1.-1.-2.-1..-1..1.-3.-3.-2..0.-2

R. H. Hardin, Table of n, a(n) for n = 1..9

A195215 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 3 or less and triangles differing by a constant counted only once.

Original entry on oeis.org

1, 37, 5453, 3395245, 9032683465, 103565705397639, 5137554141710397313, 1105495964728300359472149, 1033611648181176182212080096433, 4204060119607140359160709072654583501

Offset: 1

R. H. Hardin Sep 13 2011

nonn

Column 3 of A195220

			Some solutions for n=5
...0...............0...............0...............0...............0
..-3.-1............3..0...........-1.-1............0..3............2..1
..-1.-2.-2.........2..1..3.........2.-1.-4.........1..2..4.........3..3..4
...1..0..0.-3.....-1..2..3..2.....-1.-1.-2.-5......3..3..1..1......3..1..3..5
...3..0.-2.-1..0...1..0..2..1..0...0.-2.-2.-2.-3...2..0..1..2.-1...2..0..3..5..7

A195216 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 4 or less and triangles differing by a constant counted only once.

Original entry on oeis.org

1, 61, 18903, 31640829, 289301569283, 14572563308953245, 4059527861103273129317, 6270118871204192400242141667, 53785756580019628016324622812810239

Offset: 1

R. H. Hardin Sep 13 2011

nonn

Column 4 of A195220

			Some solutions for n=5
...0...............0...............0...............0...............0
..-4.-3...........-4.-4............4..4............3..1............2..1
..-6.-2.-5........-7.-3..0.........2..1..3.........3.-1.-1........-1..1..1
..-5.-5.-2.-4.....-4.-4.-1.-2......3..5..5..4......1..3..0.-4.....-3.-3.-1.-1
..-6.-4.-4.-5.-7..-5.-5.-2.-4.-3...2..2..1..5..8...2..3..0.-2.-2..-3.-1.-1..0..0

A195217 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 5 or less and triangles differing by a constant counted only once.

Original entry on oeis.org

1, 91, 51205, 189677411, 4677360495205, 774355028021195459, 863969905606597340602967, 6512935591253221256126051469607

Offset: 1

R. H. Hardin Sep 13 2011

nonn

Column 5 of A195220

			Some solutions for n=5
...0...............0...............0...............0...............0
...3..3............2.-2...........-1.-2...........-3..0............4..1
...2..0..4.........0..3.-1.........1..3.-1.........0..2..3.........6..2..5
..-2..0..2..1......1.-2.-1.-5......5..2..2.-2.....-3.-2.-2.-1......3..5..6..8
...0..3..0.-1..3...0.-1.-5..0.-3...3..1.-2.-3..2...1..0..0.-1.-3...6..5..6..3..3

cp's OEIS Frontend

A195213 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 1 or less and triangles differing by a constant counted only once.

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A195214 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 2 or less and triangles differing by a constant counted only once.

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A195221 Number of lower triangles of a 3 X 3 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

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A195222 Number of lower triangles of a 4 X 4 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

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A195223 Number of lower triangles of a 5 X 5 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

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A195224 Number of lower triangles of a 6 X 6 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

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A195225 Number of lower triangles of a 7X7 integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by n or less and triangles differing by a constant counted only once.

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A195215 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 3 or less and triangles differing by a constant counted only once.

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A195216 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 4 or less and triangles differing by a constant counted only once.

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A195217 Number of lower triangles of an n X n integer array with each element differing from all of its diagonal, vertical, antidiagonal and horizontal neighbors by 5 or less and triangles differing by a constant counted only once.

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