A252820 -id:A252820 - cp's OEIS frontend

A252814 Number of n X 2 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

2, 6, 17, 40, 81, 147, 246, 387, 580, 836, 1167, 1586, 2107, 2745, 3516, 4437, 5526, 6802, 8285, 9996, 11957, 14191, 16722, 19575, 22776, 26352, 30331, 34742, 39615, 44981, 50872, 57321, 64362, 72030, 80361, 89392, 99161, 109707, 121070, 133291

Offset: 1

R. H. Hardin, Dec 22 2014

nonn

			Some solutions for n=4:
..0..0....0..1....0..1....0..1....0..1....0..1....0..1....0..0....0..0....0..1
..0..1....0..1....1..2....1..2....0..1....0..1....1..2....1..1....1..1....1..2
..1..1....1..1....2..3....2..3....1..2....1..1....2..2....1..2....2..2....2..2
..2..2....1..2....2..3....3..4....2..2....2..2....3..3....1..2....2..3....2..2

R. H. Hardin, Table of n, a(n) for n = 1..210
W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.

Column 2 of A252820.

Empirical: a(n) = (1/24)*n^4 + (5/12)*n^3 - (1/24)*n^2 + (7/12)*n + 1.
Conjectures from Colin Barker, Dec 06 2018: (Start)
G.f.: x*(2 - 4*x + 7*x^2 - 5*x^3 + 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)

A252813 Number of n X n nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

1, 6, 63, 684, 7564, 86724, 1033761, 12755742, 161986236, 2106102800, 27918827562, 376114207132, 5136290930902, 70961672284626, 990271326335611, 13940676825684808, 197768062909075426, 2824828311338844486

Offset: 1

R. H. Hardin, Dec 22 2014

nonn

Diagonal of A252820

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

R. H. Hardin, Table of n, a(n) for n = 1..40

A252815 Number of n X 3 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

4, 17, 63, 187, 468, 1032, 2067, 3840, 6716, 11179, 17855, 27537, 41212, 60090, 85635, 119598, 164052, 221429, 294559, 386711, 501636, 643612, 817491, 1028748, 1283532, 1588719, 1951967, 2381773, 2887532, 3479598, 4169347, 4969242, 5892900, 6955161

Offset: 1

R. H. Hardin, Dec 22 2014

nonn

			Some solutions for n=4:
..0..0..1....0..0..0....0..1..1....0..0..1....0..1..2....0..1..2....0..1..1
..0..0..1....1..1..1....1..2..2....1..1..2....1..2..2....1..2..3....0..1..1
..1..1..2....1..1..2....2..2..3....1..2..2....2..3..3....1..2..3....1..1..2
..1..2..3....2..2..3....2..3..4....1..2..3....2..3..3....1..2..3....2..2..3

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

Column 3 of A252820.

Empirical: a(n) = (1/360)*n^6 + (1/20)*n^5 + (5/18)*n^4 + (1/2)*n^3 + (439/360)*n^2 - (1/20)*n + 2.
Conjectures from Colin Barker, Dec 06 2018: (Start)
G.f.: x*(4 - 11*x + 28*x^2 - 37*x^3 + 27*x^4 - 11*x^5 + 2*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)

A252816 Number of n X 4 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

7, 40, 187, 684, 2078, 5490, 13015, 28299, 57338, 109549, 199168, 347035, 582831, 947837, 1498290, 2309416, 3480225, 5139158, 7450681, 10622926, 14916484, 20654460, 28233905, 38138745, 50954332, 67383747, 88265990, 114596197

Offset: 1

R. H. Hardin, Dec 22 2014

nonn

			Some solutions for n=4:
..0..0..1..2....0..1..1..1....0..1..2..3....0..0..1..2....0..0..1..1
..1..1..2..3....0..1..1..2....1..2..3..3....1..1..1..2....1..1..1..2
..1..1..2..3....1..2..2..3....1..2..3..3....2..2..2..3....1..2..2..3
..1..2..3..4....1..2..3..4....2..3..3..4....2..3..3..4....1..2..3..4

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

Column 4 of A252820.

Empirical: a(n) = (1/8064)*n^8 + (1/288)*n^7 + (103/2880)*n^6 + (17/90)*n^5 + (703/1152)*n^4 + (451/288)*n^3 + (2869/3360)*n^2 + (209/120)*n + 2.
Conjectures from Colin Barker, Dec 06 2018: (Start)
G.f.: x*(7 - 23*x + 79*x^2 - 147*x^3 + 176*x^4 - 138*x^5 + 67*x^6 - 18*x^7 + 2*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)

A252817 Number of n X 5 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

11, 81, 468, 2078, 7564, 23664, 65711, 165685, 385736, 839799, 1726761, 3379640, 6336411, 11439478, 19972358, 33843927, 55832593, 89905021, 141626554, 218683266, 331538662, 494251420, 725484265, 1049738089, 1498849798

Offset: 1

R. H. Hardin, Dec 22 2014

nonn

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

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

Column 5 of A252820.

Empirical: a(n) = (1/259200)*n^10 + (1/6480)*n^9 + (149/60480)*n^8 + (163/7560)*n^7 + (10411/86400)*n^6 + (209/432)*n^5 + (17977/12960)*n^4 + (6043/3240)*n^3 + (37673/12600)*n^2 + (1423/1260)*n + 3.
Conjectures from Colin Barker, Dec 06 2018: (Start)
G.f.: x*(11 - 40*x + 182*x^2 - 430*x^3 + 711*x^4 - 822*x^5 + 657*x^6 - 360*x^7 + 131*x^8 - 29*x^9 + 3*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)

A252818 Number of n X 6 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

16, 147, 1032, 5490, 23664, 86724, 279300, 809349, 2147638, 5289321, 12219498, 26705926, 55598468, 110890224, 212893984, 395014703, 710759820, 1243830839, 2122388828, 3538891338, 5777267504, 9249641452, 14545342212, 22495562705

Offset: 1

R. H. Hardin, Dec 22 2014

nonn

Column 6 of A252820.

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

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

Cf. A252820.

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 + (319693/1036800)*n^6 + (34283/34560)*n^5 + (59941/28800)*n^4 + (482543/129600)*n^3 + (720359/277200)*n^2 + (44551/13860)*n + 3.

A252819 Number of nX7 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

Original entry on oeis.org

22, 246, 2067, 13015, 65711, 279300, 1033761, 3414257, 10248688, 28359679, 73160199, 177550620, 408377425, 895696589, 1883020893, 3810974274, 7452779978, 14128342036, 26035049083, 46748705613, 81968236755, 140604294490

Offset: 1

R. H. Hardin, Dec 22 2014

nonn

Column 7 of A252820

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

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 + (153973/907200)*n^7 + (4462807/7257600)*n^6 + (239717/145152)*n^5 + (50459/14400)*n^4 + (1240207/302400)*n^3 + (269664683/50450400)*n^2 + (308941/120120)*n + 4

cp's OEIS Frontend

A252814 Number of n X 2 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

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A252813 Number of n X n nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

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A252815 Number of n X 3 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

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A252816 Number of n X 4 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

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A252817 Number of n X 5 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

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A252818 Number of n X 6 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

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A252819 Number of nX7 nonnegative integer arrays with upper left 0 and every value within 2 of its city block distance from the upper left and every value increasing by 0 or 1 with every step right or down.

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