A208637 -id:A208637 - cp's OEIS frontend

A208638 Number of 3 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

4, 13, 32, 71, 150, 309, 628, 1267, 2546, 5105, 10224, 20463, 40942, 81901, 163820, 327659, 655338, 1310697, 2621416, 5242855, 10485734, 20971493, 41943012, 83886051, 167772130, 335544289, 671088608, 1342177247, 2684354526, 5368709085

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 3 of A208637. Possibly row 4 of the convolution array A213568. - Clark Kimberling, Jun 20 2012
From Noah Carey, Aug 31 2021: (Start)
Conjecture: a(n) is equal to half the sum along the edges of (centered, height 2, width n, starting at line n+1) rectangles in Pascal's triangle, as shown here for n=3 (not including the terms inside the rectangles):
                1
              1   1
            1   2   1     a(3) = (4+6+4 + 15+20+15)/2
          1   3   3   1
        1   4---6---4   1
      1   5 |       | 5   1
    1   6  15--20--15   6   1
  1   7  21  35  35  20   7   1 (End)

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

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

Cf. A208637.

Empirical: a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
Conjectures from Colin Barker, Mar 07 2018: (Start)
G.f.: x*(4 - 3*x) / ((1 - x)^2*(1 - 2*x)).
a(n) = 5*2^n - n - 5.
(End)

A208632 Number of n X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

1, 5, 32, 225, 1652, 12404, 94320, 722733, 5565560, 43005391, 333132384, 2585476348, 20097106512, 156418938680, 1218807197408, 9506457334965, 74216779041840, 579903491112365, 4534776883051800, 35488119754971345

Offset: 1

R. H. Hardin Feb 29 2012

nonn

Diagonal of A208637

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

R. H. Hardin, Table of n, a(n) for n = 1..27

A208633 Number of n X 4 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

8, 23, 71, 225, 722, 2331, 7548, 24476, 79423, 257807, 836973, 2717446, 8823199, 28648292, 93019712, 302032007, 980690055, 3184277945, 10339281402, 33571429395, 109005736508, 353939387572, 1149233948207, 3731539152111

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Column 4 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3).

Empirical g.f.: x*(8 - 17*x + 4*x^2) / (1 - 5*x + 6*x^2 - x^3). - Colin Barker, Jul 05 2018

A208634 Number of n X 5 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

16, 47, 150, 494, 1652, 5572, 18888, 64216, 218704, 745616, 2543520, 8679776, 29625920, 101131840, 345250944, 1178690944, 4024163584, 13739075840, 46907582976, 160151393792, 546788836352, 1866849412096, 6373813684224

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Column 5 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
Conjectures from Colin Barker, Jul 05 2018: (Start)
G.f.: x*(16 - 49*x + 28*x^2) / ((1 - 2*x)*(1 - 4*x + 2*x^2)).
a(n) = (3*2^(1+n) + (11-2*sqrt(2))*(2-sqrt(2))^n + (2+sqrt(2))^n*(11+2*sqrt(2))) / 4.
(End)

A208635 Number of n X 6 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

32, 95, 309, 1042, 3577, 12404, 43284, 151656, 532795, 1875161, 6607478, 23301225, 82215220, 290187784, 1024490960, 3617470935, 12774594765, 45114821146, 159335044937, 562751474084, 1987608261228, 7020221345560, 24795605196403

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Column 6 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 7*a(n-1) - 15*a(n-2) + 10*a(n-3) - a(n-4).

Empirical g.f.: x*(32 - 129*x + 124*x^2 - 16*x^3) / ((1 - x)*(1 - 6*x + 9*x^2 - x^3)). - Colin Barker, Jul 05 2018

A208636 Number of n X 7 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

64, 191, 628, 2149, 7504, 26508, 94320, 337227, 1209736, 4349981, 15668332, 56505640, 203961088, 736686999, 2662084284, 9622940853, 34793691400, 125826023972, 455089068016, 1646125164451, 5954682909712, 21541486064685

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Column 7 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 8*a(n-1) - 21*a(n-2) + 20*a(n-3) - 5*a(n-4).

Empirical g.f.: x*(64 - 321*x + 444*x^2 - 144*x^3) / ((1 - 3*x + x^2)*(1 - 5*x + 5*x^2)). - Colin Barker, Jul 05 2018

A208639 Number of 4 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

8, 34, 95, 225, 494, 1042, 2149, 4375, 8840, 17784, 35687, 71509, 143170, 286510, 573209, 1146627, 2293484, 4587220, 9174715, 18349729, 36699782, 73399914, 146800205, 293600815, 587202064, 1174404592, 2348809679, 4697619885

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 4 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>5.
Conjectures from Colin Barker, Jul 05 2018: (Start)
G.f.: x*(8 - 6*x - 3*x^2 + 2*x^4) / ((1 - x)^3*(1 - 2*x)).
a(n) = (-42 + 35*2^n - 13*n - n^2) / 2 for n>1.
(End)

A208640 Number of 5 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

16, 89, 284, 722, 1652, 3577, 7504, 15448, 31440, 63543, 127884, 256718, 514556, 1030421, 2062360, 4126468, 8254936, 16512147, 33026868, 66056634, 132116516, 264236657, 528477344, 1056959152, 2113923232, 4227851887, 8455709724

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 5 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5) for n>7.
Conjectures from Colin Barker, Jul 05 2018: (Start)
G.f.: x*(16 - 7*x - 26*x^2 + 8*x^3 + 16*x^4 - 2*x^5 - 4*x^6) / ((1 - x)^4*(1 - 2*x)).
a(n) = -84 + 63*2^n - (191*n)/6 - 4*n^2 - n^3/6 for n>2.
(End)

A208641 Number of 6 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

32, 233, 851, 2331, 5572, 12404, 26508, 55260, 113427, 230559, 465773, 937321, 1881726, 3772054, 7554458, 15121266, 30257157, 60531513, 121083123, 242189591, 484406152, 968843304, 1937722072, 3875484536, 7751014887, 15502081539

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 6 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 7*a(n-1) - 20*a(n-2) + 30*a(n-3) - 25*a(n-4) + 11*a(n-5) - 2*a(n-6) for n>9.
Conjectures from Colin Barker, Jul 05 2018: (Start)
G.f.: x*(32 + 9*x - 140*x^2 + 74*x^3 + 85*x^4 - 37*x^5 - 34*x^6 + 4*x^7 + 8*x^8) / ((1 - x)^5*(1 - 2*x)).
a(n) = (792*(7*2^n-10) - 3382*n - 539*n^2 - 38*n^3 - n^4) / 24 for n>3.
(End)

A208642 Number of 7 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

64, 610, 2552, 7548, 18888, 43284, 94320, 199299, 412962, 844943, 1714680, 3461238, 6962956, 13976742, 28016664, 56111133, 112317270, 224749641, 449637728, 899440872, 1799078160, 3598388200, 7197048672, 14394415431, 28789200714

Offset: 1

R. H. Hardin, Feb 29 2012

nonn

Row 7 of A208637.

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

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

Cf. A208637.

Empirical: a(n) = 8*a(n-1) - 27*a(n-2) + 50*a(n-3) - 55*a(n-4) + 36*a(n-5) - 13*a(n-6) + 2*a(n-7) for n>11.
Conjectures from Colin Barker, Jul 05 2018: (Start)
G.f.: x*(64 + 98*x - 600*x^2 + 402*x^3 + 428*x^4 - 378*x^5 - 144*x^6 + 77*x^7 + 78*x^8 - 8*x^9 - 16*x^10) / ((1 - x)^6*(1 - 2*x)).
a(n) = (51480*(-3+2^(1+n)) - 71394*n - 13145*n^2 - 1205*n^3 - 55*n^4 - n^5) / 120 for n>4.
(End)

cp's OEIS Frontend

A208638 Number of 3 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208632 Number of n X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208633 Number of n X 4 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208634 Number of n X 5 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208635 Number of n X 6 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208636 Number of n X 7 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208639 Number of 4 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208640 Number of 5 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A208641 Number of 6 X n 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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