A222769 -id:A222769 - cp's OEIS frontend

A222763 Number of n X 2 0..1 arrays with exactly floor(nX2/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

1, 0, 3, 4, 20, 48, 175, 512, 1719, 5400, 17776, 57420, 188656, 617176, 2033175, 6697744, 22139780, 73262232, 242931321, 806516560, 2681475048, 8925158440, 29740390672, 99196158144, 331163178475, 1106489052968, 3699881730900, 12380449027324, 41454579098852

Offset: 0

R. H. Hardin, Mar 05 2013

nonn

From Gus Wiseman, Oct 05 2022: (Start)
Conjecture: Also the number of integer compositions of 2n + 1 with the same length as reverse-alternating sum. Here, the reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i. For example, the a(4) = 20 compositions are:
  (135)    (234)    (333)    (432)    (531)
  (11115)  (21114)  (31113)  (41112)  (51111)
  (11214)  (21213)  (31212)  (41211)
  (11313)  (21312)  (31311)
  (11412)  (21411)
  (11511)
This is the odd-indexed version of A357182, and the corresponding unordered count (partitions) is A357488.
(End)

			All solutions for n=3
..0..1....0..0....0..0....0..0
..0..0....0..0....0..1....1..0
..0..0....1..0....0..0....0..0

Alois P. Heinz, Table of n, a(n) for n = 0..1881 (terms n = 1..210 from R. H. Hardin)

Column k = 2 of A222769.

Cf. A046736, A105422.

a:= proc(n) option remember; `if`(n<3, [1, 0, 3][n+1],
      (4*(n-1)*(74*n^2-153*n+73)*a(n-1) +8*(2*n-3)*
      (74*n^2-153*n+70)*a(n-2) -2*(37*n-21)*(2*n-5)*
      (n-1)*a(n-3))/(5*(37*n-58)*n*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 24 2024

a(n) = A105422(2n,n). - Alois P. Heinz, Sep 24 2024

a(n) = (n+1)*A046736(n+2). - Mark van Hoeij, Nov 29 2024

a(0)=1 prepended by Alois P. Heinz, Sep 24 2024

A222764 Number of nX3 0..1 arrays with exactly floor(nX3/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

0, 4, 13, 96, 301, 2242, 7575, 55958, 196442, 1448568, 5188656, 38338122, 138866688, 1029494580, 3754449699, 27929878716, 102325331440, 763577070328, 2806916894109, 21001612120424, 77405327921654, 580452846750048

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Column 3 of A222769

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

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

A222765 Number of nX4 0..1 arrays with exactly floor(nX4/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

3, 11, 66, 465, 3431, 25772, 196185, 1508369, 11685222, 91068653, 713197411, 5607868082, 44244333098, 350085423285, 2776997948510, 22076166728109, 175834747565068, 1402892278684566, 11209941707509701, 89696333927946099

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Column 4 of A222769

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

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

A222766 Number of nX5 0..1 arrays with exactly floor(nX5/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

4, 18, 158, 2608, 19766, 333978, 2645385, 45830457, 370518626, 6504289924, 53224723750, 942396671062, 7774316625479, 138492173604656, 1149166452333809, 20564313304270211, 171388475477573948, 3077759439903425070

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Column 5 of A222769

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

R. H. Hardin, Table of n, a(n) for n = 1..205

A222767 Number of nX6 0..1 arrays with exactly floor(nX6/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

4, 73, 934, 15077, 256546, 4479721, 79612578, 1432512703, 26011740182, 475591230143, 8742929996332, 161434491485598, 2991759796200596, 55616288265939185, 1036645299974444114, 19366819201445345894

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Column 6 of A222769

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

R. H. Hardin, Table of n, a(n) for n = 1..152

A222768 Number of nX7 0..1 arrays with exactly floor(nX7/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

5, 152, 2446, 89811, 1570588, 62017670, 1139211468, 46316300783, 872037778022, 36016907441620, 687733392122744, 28680675666813436, 552611942013475560, 23197864298620002960, 449800925803708450740, 18972874575715876051106

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Column 7 of A222769

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

R. H. Hardin, Table of n, a(n) for n = 1..105

A222770 Number of 2Xn 0..1 arrays with exactly floor(2Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

0, 3, 4, 11, 18, 73, 152, 470, 1022, 3361, 7840, 23386, 57626, 172387, 434280, 1257025, 3275034, 9399647, 24776572, 70325187, 188540168, 531573662, 1436226680, 4032772569, 10987025446, 30742228871, 84180549328, 235159618106

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Row 2 of A222769

			All solutions for n=3
..0..0..0....0..0..0....0..0..1....0..0..1
..0..1..1....1..0..0....1..1..1....0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..107

A222771 Number of 3Xn 0..1 arrays with exactly floor(3Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

0, 4, 13, 66, 158, 934, 2446, 14567, 39972, 240130, 673665, 4073862, 11545686, 70212506, 200446066, 1224207384, 3512430906, 21526789216, 62001340484, 381049009426, 1100824791540, 6780941436208, 19637801359798, 121196222602133

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Row 3 of A222769

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

R. H. Hardin, Table of n, a(n) for n = 1..48

A222772 Number of 4Xn 0..1 arrays with exactly floor(4Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

0, 20, 96, 465, 2608, 15077, 89811, 544302, 3349328, 20823047, 130535918, 823585242, 5221057054, 33231082276, 212180118775, 1358528423728, 8718262087314, 56062919553225, 361148981248207, 2330144611056838

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Row 4 of A222769

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

R. H. Hardin, Table of n, a(n) for n = 1..45

A222773 Number of 5Xn 0..1 arrays with exactly floor(5Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

Original entry on oeis.org

0, 48, 301, 3431, 19766, 256546, 1570588, 21526008, 136549058, 1916013454, 12429454430, 176824724591, 1163475046332, 16694274193142, 110869265032135, 1600019787081011, 10694449471174787, 154977656102006006

Offset: 1

R. H. Hardin Mar 05 2013

nonn

Row 5 of A222769

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

R. H. Hardin, Table of n, a(n) for n = 1..39

cp's OEIS Frontend

A222763 Number of n X 2 0..1 arrays with exactly floor(nX2/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222764 Number of nX3 0..1 arrays with exactly floor(nX3/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222765 Number of nX4 0..1 arrays with exactly floor(nX4/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222766 Number of nX5 0..1 arrays with exactly floor(nX5/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222767 Number of nX6 0..1 arrays with exactly floor(nX6/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222768 Number of nX7 0..1 arrays with exactly floor(nX7/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222770 Number of 2Xn 0..1 arrays with exactly floor(2Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222771 Number of 3Xn 0..1 arrays with exactly floor(3Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222772 Number of 4Xn 0..1 arrays with exactly floor(4Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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A222773 Number of 5Xn 0..1 arrays with exactly floor(5Xn/2) elements unequal to at least one horizontal or antidiagonal neighbor, with new values introduced in row major 0..1 order.

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