A253223 -id:A253223 - cp's OEIS frontend

A253217 Number of n X n nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

0, 0, 1, 19, 268, 3568, 47698, 649712, 9023385, 127419681, 1823918697, 26398702645, 385582981615, 5674890516295, 84060883775765, 1252066289632643, 18738613233957420, 281620474177057788, 4248088188086420832

Offset: 1

R. H. Hardin, Dec 29 2014

nonn

Diagonal of A253223.

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

R. H. Hardin, Table of n, a(n) for n = 1..37
Robert Dougherty-Bliss, Experimental Methods in Number Theory and Combinatorics, Ph. D. Dissertation, Rutgers Univ. (2024). See p. 29.
Robert Dougherty-Bliss and Manuel Kauers, Hardinian Arrays, arXiv:2309.00487 [math.CO], 2023.
Manuel Kauers and Christoph Koutschan, Guessing with Little Data, ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, July 2022, Pages 83-90.
Manuel Kauers and Christoph Koutschan, Some D-finite and Some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023.

Recurrence: 32*(1 + n)*(1 + 2*n)^2*(161046 + 465785*n + 551943*n^2 + 343020*n^3 + 117954*n^4 + 21285*n^5 + 1575*n^6)*a(n) - 8*(4443102 + 33718283*n + 105734340*n^2 + 180574335*n^3 + 186866686*n^4 + 122556360*n^5 + 51280818*n^6 + 13267683*n^7 + 1933470*n^8 + 121275*n^9)*a(n+1) + 2*(12137328 + 91378536*n + 283626704*n^2 + 478464380*n^3 + 488415476*n^4 + 315713355*n^5 + 130145646*n^6 + 33170868*n^7 + 4763070*n^8 + 294525*n^9)*a(n+2) + (10688508 + 80866406*n + 252913504*n^2 + 431097970*n^3 + 445804136*n^4 + 292620525*n^5 + 122735586*n^6 + 31877118*n^7 + 4668570*n^8 + 294525*n^9)*a(n+3) - (4877748 + 36871922*n + 114948300*n^2 + 194784258*n^3 + 199650088*n^4 + 129484209*n^5 + 53503836*n^6 + 13655808*n^7 + 1961820*n^8 + 121275*n^9)*a(n+4) + 2*(3 + n)^2*(7 + 2*n)*(2428 + 16118*n + 41382*n^2 + 52554*n^3 + 35154*n^4 + 11835*n^5 + 1575*n^6)*a(n+5) = 0. - conjectured by Manuel Kauers and Christoph Koutschan, Mar 02 2023; proved by Robert Dougherty-Bliss and Manuel Kauers

Conjecture: a(n) ~ 2^(4*n - 2) / (81 * Pi * n), based on the above recurrence - Vaclav Kotesovec, Mar 02 2023

A253218 Number of n X 3 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Original entry on oeis.org

1, 1, 1, 19, 102, 263, 504, 825, 1226, 1707, 2268, 2909, 3630, 4431, 5312, 6273, 7314, 8435, 9636, 10917, 12278, 13719, 15240, 16841, 18522, 20283, 22124, 24045, 26046, 28127, 30288, 32529, 34850, 37251, 39732, 42293, 44934, 47655, 50456, 53337, 56298

Offset: 1

R. H. Hardin, Dec 29 2014

nonn

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

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

Column 3 of A253223.

Empirical: a(n) = 40*n^2 - 279*n + 497 for n>4.
Conjectures from Colin Barker, Dec 09 2018: (Start)
G.f.: x*(1 - 2*x + x^2 + 18*x^3 + 47*x^4 + 13*x^5 + 2*x^6) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>7.
(End)

A253219 Number of n X 4 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Original entry on oeis.org

3, 9, 19, 19, 268, 1249, 3140, 5986, 9792, 14558, 20284, 26970, 34616, 43222, 52788, 63314, 74800, 87246, 100652, 115018, 130344, 146630, 163876, 182082, 201248, 221374, 242460, 264506, 287512, 311478, 336404, 362290, 389136, 416942, 445708

Offset: 1

R. H. Hardin, Dec 29 2014

nonn

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

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

Column 4 of A253223.

Empirical: a(n) = 480*n^2 - 4354*n + 10098 for n>6.
Conjectures from Colin Barker, Dec 10 2018: (Start)
G.f.: x*(3 + x^2 - 14*x^3 + 259*x^4 + 483*x^5 + 178*x^6 + 45*x^7 + 5*x^8) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>9.
(End)

A253220 Number of n X 5 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Original entry on oeis.org

6, 25, 102, 268, 268, 3568, 16028, 40238, 77063, 126673, 189083, 264293, 352303, 453113, 566723, 693133, 832343, 984353, 1149163, 1326773, 1517183, 1720393, 1936403, 2165213, 2406823, 2661233, 2928443, 3208453, 3501263, 3806873, 4125283

Offset: 1

R. H. Hardin, Dec 29 2014

nonn

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

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

Column 5 of A253223.

Empirical: a(n) = 6400*n^2 - 71990*n + 206573 for n>8.
Conjectures from Colin Barker, Dec 10 2018: (Start)
G.f.: x*(6 + 7*x + 45*x^2 + 31*x^3 - 255*x^4 + 3466*x^5 + 5860*x^6 + 2590*x^7 + 865*x^8 + 170*x^9 + 15*x^10) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>11.
(End)

A253221 Number of n X 6 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Original entry on oeis.org

10, 49, 263, 1249, 3568, 3568, 47698, 213155, 538444, 1039060, 1719174, 2579462, 3619974, 4840710, 6241670, 7822854, 9584262, 11525894, 13647750, 15949830, 18432134, 21094662, 23937414, 26960390, 30163590, 33547014, 37110662, 40854534

Offset: 1

R. H. Hardin, Dec 29 2014

nonn

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

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

Column 6 of A253223.

Empirical: a(n) = 90112*n^2 - 1212288*n + 4150790 for n > 10.
Conjectures from Colin Barker, Dec 10 2018: (Start)
G.f.: x*(10 + 19*x + 146*x^2 + 597*x^3 + 561*x^4 - 3652*x^5 + 46449*x^6 + 77197*x^7 + 38505*x^8 + 15495*x^9 + 4171*x^10 + 676*x^11 + 50*x^12) / (1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 13.
(End)

A253222 Number of n X 7 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Original entry on oeis.org

15, 81, 504, 3140, 16028, 47698, 47698, 649712, 2913793, 7415837, 14420585, 24015795, 36221327, 51039931, 68471783, 88516883, 111175231, 136446827, 164331671, 194829763, 227941103, 263665691, 302003527, 342954611, 386518943

Offset: 1

R. H. Hardin, Dec 29 2014

nonn

Column 7 of A253223.

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

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

Cf. A253223.

Empirical: a(n) = 1306624*n^2 - 20460244*n + 81385043 for n>12.

cp's OEIS Frontend

A253217 Number of n X n nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Views

Author

Keywords

Comments

Examples

Links

Formula

A253218 Number of n X 3 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A253219 Number of n X 4 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A253220 Number of n X 5 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A253221 Number of n X 6 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A253222 Number of n X 7 nonnegative integer arrays with upper left 0 and lower right its king-move distance away minus 2 and every value within 2 of its king move distance from the upper left and every value increasing by 0 or 1 with every step right, diagonally se or down.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula