cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A202736 Number of n X 2 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 2, 8, 10, 32, 44, 128, 186, 512, 772, 2048, 3172, 8192, 12952, 32768, 52666, 131072, 213524, 524288, 863820, 2097152, 3488872, 8388608, 14073060, 33554432, 56708264, 134217728, 228318856, 536870912, 918624304, 2147483648, 3693886906, 8589934592, 14846262964
Offset: 1

Views

Author

R. H. Hardin, Dec 23 2011

Keywords

Comments

Column 2 of A202742.
a(n) is the number of binary words of length n such that the number of 0's is not equal to the number of 1's. - Geoffrey Critzer, Dec 05 2013
Also the degree of the irreducible polynomial that defines the multifocal ellipsoid with n foci, see links. - Moritz Firsching, Aug 31 2015

Examples

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

Links

Crossrefs

Cf. A126869.

Programs

  • Magma
    I:=[2,2,8]; [n le 3 select I[n] else ((-8*n+16)*Self(n-3)+(4*n-4)*Self(n-2)+(2*n-2)*Self(n-1))/n: n in [1..40]]; // Vincenzo Librandi, Sep 01 2015
  • Maple
    seq(2^n - `if`(n::even, binomial(n,n/2), 0), n = 1 .. 30); # Robert Israel, Aug 31 2015
  • Mathematica
    f[n_]:= If[EvenQ[n],2^n-Binomial[n,n/2],2^n];Drop[Table[f[n],{n,0,34}],1] (* Geoffrey Critzer, Dec 05 2013 *)
  • Sage
    A202736 = lambda n: 2^n*(1-(((-1)^n+1)*gamma((n+1)/2))/(2*sqrt(pi)*gamma((n+2)/2)))
[A202736(n) for n in (1..32)] # Peter Luschny, Sep 10 2014

Formula

For odd n, a(n) = 2^n, for even n, a(n) = 2^n - binomial(n,n/2). - Geoffrey Critzer, Dec 05 2013
a(n) = 2^n*(1-(((-1)^n+1)*Gamma((n+1)/2))/(2*sqrt(Pi)*Gamma((n+2)/2))). - Peter Luschny, Sep 10 2014
a(n) = 2^n - A126869(n). - Peter Luschny, Sep 10 2014
From Robert Israel, Aug 31 2015: (Start)
G.f.: 1/(1-2*x) - 1/sqrt(1-4*x^2).
E.g.f.: exp(2*x) - I_0(2*x) where I_0 is a modified Bessel function.
a(n) = ((-8*n+16)*a(n-3)+(4*n-4)*a(n-2)+(2*n-2)*a(n-1))/n. (End)

A202737 Number of nX3 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 6, 38, 86, 282, 1074, 2802, 8694, 30170, 82286, 251682, 837026, 2336882, 7101030, 23060898, 65306934, 197740602, 632536014, 1808535522, 5464051326, 17297449410, 49797577110, 150233445162, 471996750978, 1365869873682, 4116478752054
Offset: 1

Views

Author

R. H. Hardin Dec 23 2011

Keywords

Comments

Column 3 of A202742

Examples

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

Links

A202738 Number of nX4 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 22, 140, 838, 5092, 29680, 183220, 1061814, 6630116, 39048472, 242647660, 1438692856, 8910239756, 53090781836, 327214611580, 1954347351734, 11998820896676, 71771002591048, 439163289580348, 2629051988484048
Offset: 1

Views

Author

R. H. Hardin Dec 23 2011

Keywords

Comments

Column 4 of A202742

Examples

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

Links

A202739 Number of n X 5 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 54, 638, 6870, 75302, 827502, 8870150, 93402390, 969837470, 10133577294, 104168885570, 1073411553030, 11008021249370, 112437630047286, 1151811769082618, 11687452535796630, 118872499759443950
Offset: 1

Views

Author

R. H. Hardin, Dec 23 2011

Keywords

Comments

Column 5 of A202742.

Examples

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

Links

Crossrefs

Cf. A202742.

A202740 Number of nX6 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 102, 3032, 62398, 1375912, 27842760, 578826040, 11600238174, 239443637912, 4806953288872, 98795215188760, 1991699775933928, 40848655904440568, 824853028164662088, 16889107635602264212, 341245428784872621534
Offset: 1

Views

Author

R. H. Hardin Dec 23 2011

Keywords

Comments

Column 6 of A202742

Examples

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

Links

A202743 Number of 2Xn 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 2, 6, 22, 54, 102, 214, 582, 1590, 3814, 8662, 20934, 53942, 137190, 335574, 815942, 2032694, 5139942, 12902614, 32059846, 79807158, 200325350, 504502230, 1267146054, 3176193590, 7975495910, 20082938838, 50611676614, 127466481334
Offset: 1

Views

Author

R. H. Hardin Dec 23 2011

Keywords

Comments

Row 2 of A202742

Examples

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

Links

A202744 Number of 3Xn 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 8, 38, 140, 638, 3032, 14258, 70136, 350594, 1767452, 9029918, 46519544, 241210598, 1258499756, 6598909298, 34748813672, 183679125026, 974136322892, 5181534476414, 27634187724392, 147730324961126, 791463055043228
Offset: 1

Views

Author

R. H. Hardin Dec 23 2011

Keywords

Comments

Row 3 of A202742

Examples

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

Links

A202741 Number of n X 7 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 214, 14258, 613950, 24620522, 967878370, 37121438930, 1401569558590, 52383342552410, 1938999772255314, 71224141762895090
Offset: 1

Views

Author

R. H. Hardin, Dec 23 2011

Keywords

Comments

Column 7 of A202742.

Examples

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

Crossrefs

Cf. A202742.

A202745 Number of 4Xn 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 10, 86, 838, 6870, 62398, 613950, 6018166, 59293518, 601267870, 6215708078, 64727555230, 679456522670, 7200537529326, 76879249259246, 825573117666246, 8913831877350462, 96733061472889806, 1054372579551732366
Offset: 1

Views

Author

R. H. Hardin Dec 23 2011

Keywords

Comments

Row 4 of A202742

Examples

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

A202746 Number of 5 X n 0..1 arrays with row sums equal and column sums unequal to adjacent columns.

Original entry on oeis.org

2, 32, 282, 5092, 75302, 1375912, 24620522, 472929952, 9181463302, 183977062692, 3737085533882
Offset: 1

Views

Author

R. H. Hardin Dec 23 2011

Keywords

Comments

Row 5 of A202742.

Examples

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

Crossrefs

Cf. A202742.
Showing 1-10 of 12 results. Next

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