A204651 -id:A204651 - cp's OEIS frontend

A204644 Number of (n+1) X 2 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

8, 16, 28, 48, 80, 132, 216, 352, 572, 928, 1504, 2436, 3944, 6384, 10332, 16720, 27056, 43780, 70840, 114624, 185468, 300096, 485568, 785668, 1271240, 2056912, 3328156, 5385072, 8713232, 14098308, 22811544, 36909856, 59721404, 96631264, 156352672, 252983940

Offset: 1

R. H. Hardin, Jan 17 2012

nonn

			Some solutions for n=5:
  0 1   0 1   0 0   0 0   0 1   0 0   1 0   0 0   0 0   0 0
  0 1   0 1   0 0   0 0   1 0   0 1   0 1   0 0   0 0   1 1
  1 0   0 1   0 0   0 0   0 1   0 1   1 0   0 1   0 0   1 1
  0 1   0 1   0 1   0 0   1 1   0 1   0 1   0 1   0 1   1 1
  0 1   1 0   1 0   0 0   1 1   0 1   1 0   0 1   1 1   1 1
  1 1   0 1   0 1   0 1   1 1   0 1   0 1   1 1   1 1   1 1

R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1) (empirical).

Column 1 of A204651.

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

From the empirical recurrence, a(n) = 4*(Fibonacci(n + 3) - 1). - Ehren Metcalfe, Apr 04 2019

A204643 Number of (n+1)X(n+1) 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

Original entry on oeis.org

8, 32, 104, 330, 1076, 3702, 13314, 49312, 185786, 706912, 2706218, 10403402, 40120324, 155122378, 601086622, 2333614100, 9075145134, 35345275932, 137846543630, 538257892350, 2104098985192, 8233430753142, 32247603713264

Offset: 1

R. H. Hardin Jan 17 2012

nonn

Diagonal of A204651

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

R. H. Hardin, Table of n, a(n) for n = 1..63

A204645 Number of (n+1) X 3 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

Original entry on oeis.org

16, 32, 56, 90, 137, 200, 283, 390, 526, 696, 906, 1162, 1471, 1840, 2277, 2790, 3388, 4080, 4876, 5786, 6821, 7992, 9311, 10790, 12442, 14280, 16318, 18570, 21051, 23776, 26761, 30022, 33576, 37440, 41632, 46170, 51073, 56360, 62051, 68166, 74726, 81752

Offset: 1

R. H. Hardin, Jan 17 2012

nonn

Column 2 of A204651.

			Some solutions for n=5:
..0..0..0....0..0..1....0..0..0....0..0..0....0..0..0....0..0..1....0..1..1
..0..0..0....0..0..1....0..0..1....0..0..1....0..0..0....0..1..1....0..1..1
..0..0..0....0..0..1....0..1..1....0..0..1....0..0..0....0..1..1....0..1..1
..0..0..0....0..1..1....0..1..1....0..0..1....0..1..1....0..1..1....0..1..1
..0..0..1....0..1..1....0..1..1....0..0..1....1..1..1....1..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

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

Cf. A204651.

Empirical: a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6).
Conjectures from Colin Barker, Jun 08 2018: (Start)
G.f.: x*(16 - 32*x + 8*x^2 + 26*x^3 - 23*x^4 + 6*x^5) / ((1 - x)^5*(1 + x)).
a(n) = (576 + 704*n + 232*n^2 + 16*n^3 + 2*n^4)/96 for n even.
a(n) = (582 + 704*n + 232*n^2 + 16*n^3 + 2*n^4)/96 for n odd.
(End)

A204646 Number of (n+1) X 4 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

Original entry on oeis.org

28, 56, 104, 178, 284, 434, 637, 908, 1259, 1708, 2270, 2966, 3814, 4838, 6059, 7504, 9197, 11168, 13444, 16058, 19040, 22426, 26249, 30548, 35359, 40724, 46682, 53278, 60554, 68558, 77335, 86936, 97409, 108808, 121184, 134594, 149092, 164738, 181589

Offset: 1

R. H. Hardin, Jan 17 2012

nonn

Column 3 of A204651.

			Some solutions for n=5:
..0..0..0..1....0..0..0..1....0..0..0..0....0..0..0..0....0..0..0..0
..1..1..1..1....0..0..0..1....0..0..1..1....0..0..0..1....0..0..0..0
..1..1..1..1....0..0..0..1....1..1..1..1....0..0..0..1....0..0..0..0
..1..1..1..1....0..0..1..1....1..1..1..1....0..0..0..1....0..0..0..1
..1..1..1..1....0..1..1..1....1..1..1..1....0..0..1..1....0..0..0..1
..1..1..1..1....1..1..1..1....1..1..1..1....0..0..1..1....1..1..1..1

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

Cf. A204651.

Empirical: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6) for n>7.
Conjectures from Colin Barker, Jun 08 2018: (Start)
G.f.: x*(28 - 56*x + 20*x^2 + 42*x^3 - 48*x^4 + 20*x^5 - 3*x^6) / ((1 - x)^5*(1 + x)).
a(n) = (256 + 400*n + 144*n^2 + 16*n^3 + 2*n^4)/32 for n>1 and even.
a(n) = (238 + 400*n + 144*n^2 + 16*n^3 + 2*n^4)/32 for n>1 and odd.
(End)

A204647 Number of (n+1) X 5 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

Original entry on oeis.org

48, 90, 178, 330, 571, 938, 1478, 2248, 3317, 4766, 6690, 9198, 12415, 16482, 21558, 27820, 35465, 44710, 55794, 68978, 84547, 102810, 124102, 148784, 177245, 209902, 247202, 289622, 337671, 391890, 452854, 521172, 597489, 682486, 776882, 881434

Offset: 1

R. H. Hardin, Jan 17 2012

nonn

Column 4 of A204651.

			Some solutions for n=5:
..0..0..0..1..1....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..0..0..0..1..1....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..0..0..1..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..0..1....0..0..0..1..1....0..0..0..0..1
..1..1..1..1..1....0..0..0..0..1....0..1..1..1..1....0..0..0..1..1
..1..1..1..1..1....0..0..0..1..0....0..1..1..1..1....0..0..1..1..1

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

Cf. A204651.

Empirical: a(n) = 5*a(n-1) -9*a(n-2) +5*a(n-3) +5*a(n-4) -9*a(n-5) +5*a(n-6) -a(n-7) for n>9.
Conjectures from Colin Barker, Jun 08 2018: (Start)
G.f.: x*(48 - 150*x + 160*x^2 + 10*x^3 - 167*x^4 + 145*x^5 - 43*x^6 - 5*x^7 + 4*x^8) / ((1 - x)^6*(1 + x)).
a(n) = (1920 + 9776*n + 3480*n^2 + 540*n^3 + 90*n^4 + 4*n^5)/480 for n>2 and even.
a(n) = (1950 + 9776*n + 3480*n^2 + 540*n^3 + 90*n^4 + 4*n^5)/480 for n>2 and odd.
(End)

A204648 Number of (n+1) X 6 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

Original entry on oeis.org

80, 137, 284, 571, 1076, 1918, 3261, 5329, 8408, 12867, 19162, 27859, 39640, 55328, 75895, 102489, 136444, 179309, 232860, 299131, 380428, 479362, 598865, 742225, 913104, 1115575, 1354142, 1633779, 1959952, 2338660, 2776459, 3280505, 3858580

Offset: 1

R. H. Hardin, Jan 17 2012

nonn

Column 5 of A204651.

			Some solutions for n=5:
..1..0..1..0..1..0....0..0..0..0..0..1....0..0..0..0..0..0....0..1..1..1..1..1
..0..1..0..1..0..1....0..0..0..0..0..1....0..0..0..0..1..1....1..1..1..1..1..1
..1..0..1..0..1..1....0..0..0..0..1..1....0..0..1..1..1..1....1..1..1..1..1..1
..0..1..0..1..1..1....0..0..0..0..1..1....0..0..1..1..1..1....1..1..1..1..1..1
..1..0..1..1..1..1....0..0..0..1..1..1....0..0..1..1..1..1....1..1..1..1..1..1
..0..1..1..1..1..1....0..0..1..1..1..1....0..0..1..1..1..1....1..1..1..1..1..1

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

Cf. A204651.

Empirical: a(n) = 6*a(n-1) -14*a(n-2) +14*a(n-3) -14*a(n-5) +14*a(n-6) -6*a(n-7) +a(n-8) for n>11.
Conjectures from Colin Barker, Jun 08 2018: (Start)
G.f.: x*(80 - 343*x + 582*x^2 - 335*x^3 - 292*x^4 + 600*x^5 - 379*x^6 + 89*x^7 - 4*x^8 + 8*x^9 - 4*x^10) / ((1 - x)^7*(1 + x)).
a(n) = (-10080 + 43776*n + 15308*n^2 + 2970*n^3 + 620*n^4 + 54*n^5 + 2*n^6)/1440 for n>3 and even.
a(n) = (-10890 + 43776*n + 15308*n^2 + 2970*n^3 + 620*n^4 + 54*n^5 + 2*n^6)/1440 for n>3 and odd.
(End)

A204649 Number of (n+1) X 7 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

Original entry on oeis.org

132, 200, 434, 938, 1918, 3702, 6780, 11868, 19969, 32450, 51134, 78404, 117324, 171774, 246604, 347804, 482695, 660138, 890766, 1187236, 1564506, 2040134, 2634604, 3371676, 4278765, 5387346, 6733390, 8357828, 10307048, 12633422

Offset: 1

R. H. Hardin, Jan 17 2012

nonn

Column 6 of A204651.

			Some solutions for n=5:
..0..0..0..1..1..1..1....0..0..0..0..0..0..1....0..0..0..0..0..0..0
..0..0..1..1..1..1..1....0..0..0..0..0..0..1....0..0..0..0..0..1..1
..0..0..1..1..1..1..1....0..0..0..0..0..0..1....0..0..1..1..1..1..1
..0..0..1..1..1..1..1....0..0..0..0..1..1..1....0..0..1..1..1..1..1
..0..0..1..1..1..1..1....0..0..0..1..1..1..1....0..1..1..1..1..1..1
..0..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

Cf. A204651.

Empirical: a(n) = 7*a(n-1) -20*a(n-2) +28*a(n-3) -14*a(n-4) -14*a(n-5) +28*a(n-6) -20*a(n-7) +7*a(n-8) -a(n-9) for n>13.

Empirical g.f.: x*(132 - 724*x + 1674*x^2 - 1796*x^3 + 280*x^4 + 1532*x^5 - 1858*x^6 + 992*x^7 - 255*x^8 + 15*x^9 + 26*x^10 - 22*x^11 + 6*x^12) / ((1 - x)^8*(1 + x)). - Colin Barker, Jun 08 2018

A204650 Number of (n+1) X 8 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

Original entry on oeis.org

216, 283, 637, 1478, 3261, 6780, 13314, 24862, 44426, 76378, 126906, 204583, 321038, 491781, 737163, 1083525, 1564519, 2222657, 3111073, 4295556, 5856841, 7893218, 10523448, 13890048, 18162936, 23543500, 30269084, 38617957, 48914760, 61536499

Offset: 1

R. H. Hardin, Jan 17 2012

nonn

Column 7 of A204651.

			Some solutions for n=5:
  1 0 1 0 1 0 0 1       0 0 0 0 0 0 0 0       0 0 0 0 0 0 0 1
  0 1 0 1 0 1 1 1       0 0 0 0 0 0 0 1       0 0 0 0 0 0 0 1
  1 0 1 0 1 1 1 1       0 0 0 0 0 0 0 1       0 0 0 0 0 0 1 1
  0 1 0 1 1 1 1 1       0 0 0 1 1 1 1 1       0 0 0 0 0 1 1 1
  1 0 1 1 1 1 1 1       1 1 1 1 1 1 1 1       0 0 0 0 0 1 1 1
  0 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

Cf. A204651.

Empirical: a(n) = 8*a(n-1) - 27*a(n-2) + 48*a(n-3) - 42*a(n-4) + 42*a(n-6) - 48*a(n-7) + 27*a(n-8) - 8*a(n-9) + a(n-10) for n > 15.

Empirical g.f.: x*(216 - 1445*x + 4205*x^2 - 6345*x^3 + 4124*x^4 + 1908*x^5 - 6141*x^6 + 5440*x^7 - 2472*x^8 + 519*x^9 + 27*x^10 - 28*x^11 - 24*x^12 + 24*x^13 - 6*x^14) / ((1 - x)^9*(1 + x)). - Colin Barker, Jun 08 2018

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A204644 Number of (n+1) X 2 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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A204643 Number of (n+1)X(n+1) 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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A204645 Number of (n+1) X 3 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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A204646 Number of (n+1) X 4 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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A204647 Number of (n+1) X 5 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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A204648 Number of (n+1) X 6 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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A204649 Number of (n+1) X 7 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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A204650 Number of (n+1) X 8 0..1 arrays with column and row pair sums b(i,j)=a(i,j)+a(i,j-1) and c(i,j)=a(i,j)+a(i-1,j) nondecreasing in column and row directions, respectively.

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