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 91 results. Next

A069361 Number of 3 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 17, 197, 1985, 18621, 167337, 1461797, 12519345, 105683341, 882516857, 7308428597, 60131384705, 492202181661, 4012347269577, 32599584662597, 264152863210065, 2135714594033581, 17236446198921497, 138901692341235797, 1117982939085627425, 8989229069675479101
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Examples

			The 17 binary arrays for n=2:
01 10 01 10 01 10 01 10 01 10 11 11 11 11 11 11 11
01 10 01 10 11 11 11 11 11 11 01 10 01 01 11 11 11
01 10 11 11 01 10 10 01 11 11 01 10 11 11 01 10 11 - _R. J. Mathar_, Jun 21 2023

Links

Crossrefs

Row 3 of A359576.
Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

Programs

  • Mathematica
    CoefficientList[Series[(-2 z - 1)/(16 z^3 - 58 z^2 + 15 z - 1), {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 24 2011 *)
  • PARI
    x='x+O('x^30); Vec(x*(1+2*x)/((1-8*x)*(2*x^2-7*x+1))) \\ G. C. Greubel, Apr 22 2018

Formula

G.f.: x*(1+2*x)/((1-8*x)*(2*x^2-7*x+1)). - Vladeta Jovovic, Jul 02 2003
From Maksym Voznyy (voznyy(AT)mail.ru), Jul 25 2008: (Start)
a(n) = 15*a(n-1) - 58*a(n-2) + 16*a(n-3), where a(1)=1, a(2)=17, a(3)=197;
a(n) = 8^n + 1/sqrt(41)*4^(n+1)*((7+sqrt(41))^(-(n+1)) - (7-sqrt(41))^(-(n+1))). (End)
a(n) = 8^n - A186446(n). - R. J. Mathar, Jan 27 2020

A069429 Half the number of 3 X n binary arrays with no path of adjacent 1's or adjacent 0's from top row to bottom row.

Original entry on oeis.org

3, 16, 84, 440, 2304, 12064, 63168, 330752, 1731840, 9068032, 47480832, 248612864, 1301753856, 6816071680, 35689414656, 186872201216, 978475548672, 5123364487168, 26826284728320, 140464250421248, 735480363614208, 3851025180000256, 20164229625544704, 105581277033267200
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Examples

			From _Andrew Howroyd_, Oct 27 2020: (Start)
Some of the 2*a(2) = 32 arrays are:
  0 0   0 0   0 0   0 1   0 1   0 0   0 1
  0 0   0 1   1 1   1 0   1 0   1 1   1 0
  1 1   1 1   1 1   1 1   0 1   0 0   1 1
(End)

Links

Crossrefs

Cf. 2 X n A000079, n X 1 A000225, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.
Cf. A084326.

Programs

Formula

Empirical G.f.: x*(3-2*x)/(1-6*x+4*x^2). - Colin Barker, Feb 22 2012
Empirical: a(n) = 3*A084326(n) - 2*A084326(n-1). - R. J. Mathar, Nov 09 2018
From Andrew Howroyd, Oct 27 2020: (Start)
The above conjectures are true and follow from formulas given in A069361 and A069396.
a(n) = (8^n)/2 - A069361(n) + A069396(n).
a(n) = 2^(n-1)*Fibonacci(2*n+2) = A084326(n+1)/2. (End)

Extensions

Terms a(21) and beyond from Andrew Howroyd, Oct 27 2020

A069395 Number of n X 20 binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1048575, 1096024843375, 1117982939085627425, 1092719640470296684383473
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Crossrefs

Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A069452 Half the number of 7 X n binary arrays with no path of adjacent 1's or adjacent 0's from top to bottom or side to side.

Original entry on oeis.org

63, 56757, 18772467, 3912171001
Offset: 2

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Crossrefs

Cf. 2 X n A000225, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A069396 Half the number of 3 X n binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

1, 25, 377, 4541, 48329, 476389, 4461489, 40306317, 354713977, 3060942133, 26020259201, 218626028573, 1820140085705, 15043088032837, 123602247055953, 1010793162739629, 8234370308667673, 66870924588036181
Offset: 2

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

Cf. 1 X n A000225, 2 X n A016269, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x+1)/(4*x^2-6*x+1))); // G. C. Greubel, Apr 22 2018
  • Mathematica
    Drop[CoefficientList[Series[x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x + 1)/(4*x^2 - 6*x + 1), {x, 0, 50}], x], 2] (* G. C. Greubel, Apr 22 2018 *)
  • PARI
    x='x+O('x^30); Vec(x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x+1)/(4*x^2 -6*x+1)) \\ G. C. Greubel, Apr 22 2018

Formula

G.f.: x^2*(2*x+1)^2/(1-8*x)/(2*x^2-7*x+1)/(4*x^2-6*x+1). - Vladeta Jovovic, Jul 02 2003
2*a(n) = 8^n+A084326(n+1) -2*A186446(n). - R. J. Mathar, May 09 2023

A069416 Half the number of n X 16 binary arrays with a path of adjacent 1's and a path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

32767, 2104469695, 123602247055953, 6475978445076745163
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Crossrefs

Cf. 1 X n A000225, 2 X n A016269, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A069417 Number of 3 X n binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

1, 15, 147, 1231, 9539, 70679, 509019, 3596367, 25070707, 173088903, 1186544331, 8090866303, 54950124515, 372067098167, 2513408596923, 16948369098159, 114128268554323, 767705581586151, 5159843165163435, 34657637020377055, 232672006452068291, 1561421588852637335
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Examples

			From _Andrew Howroyd_, Oct 27 2020: (Start)
Some of the a(2) = 15 arrays are:
  1 0   1 0   1 0   1 1   1 0
  1 1   1 0   1 1   1 1   1 1
  1 0   1 1   1 1   1 1   0 1
(End)

Links

Crossrefs

Cf. 2 X n A001047, n X 2 A034182, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

Programs

  • Mathematica
    LinearRecurrence[{13, -48, 40, -8}, {1, 15, 147, 1231}, 25] (* Paolo Xausa, Feb 08 2024 *)
  • PARI
    Vec((1 + 2*x)/((1 - 7*x + 2*x^2)*(1 - 6*x + 4*x^2)) + O(x^25)) \\ Andrew Howroyd, Oct 27 2020

Formula

From Andrew Howroyd, Oct 27 2020: (Start)
a(n) = A069361(n) - 2*A069396(n).
a(n) = 13*a(n-1) - 48*a(n-2) + 40*a(n-3) - 8*a(n-4) for n > 4.
G.f.: x*(1 + 2*x)/((1 - 7*x + 2*x^2)*(1 - 6*x + 4*x^2)).
(End)

Extensions

Terms a(12) and beyond from Andrew Howroyd, Oct 27 2020

A069428 Number of n X 8 binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row.

Original entry on oeis.org

1, 6305, 3596367, 1201461339
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Crossrefs

Cf. 2 X n A001047, n X 2 A034182, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A069447 Half the number of n X 9 binary arrays with no path of adjacent 1's or adjacent 0's from top row to bottom row.

Original entry on oeis.org

256, 1731840, 3334295986, 3616567402784
Offset: 2

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Crossrefs

Cf. 2 X n A000079, n X 1 A000225, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

A069362 Number of 4 X n binary arrays with a path of adjacent 1's from top row to bottom row.

Original entry on oeis.org

1, 41, 1041, 22193, 433809, 8057905, 144769425, 2541013617, 43843180113, 746691527217, 12588144461329, 210502738714097, 3497001564166609, 57781030561348017, 950437243856526737, 15574913193760097649, 254416775893204873553, 4144677558181255455025
Offset: 1

Views

Author

R. H. Hardin, Mar 22 2002

Keywords

Links

Crossrefs

Row 4 of A359576.
Cf. 1 X n A000225, 2 X n A005061, n X 2 A001333, vertical path of 1 A069361-A069395, vertical paths of 0+1 A069396-A069416, vertical path of 1 not 0 A069417-A069428, no vertical paths A069429-A069447, no horizontal or vertical paths A069448-A069452.

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+6*x-16*x^2-8*x^3)/((1-16*x)*(1-19*x+ 74*x^2 -80*x^3-8*x^4)))); // G. C. Greubel, Apr 22 2018
  • Mathematica
    Rest[CoefficientList[Series[x*(1+6*x-16*x^2-8*x^3)/((1-16*x)*(1-19*x+ 74*x^2 -80*x^3-8*x^4)), {x,0,50}],x]] (* G. C. Greubel, Apr 22 2018 *)
LinearRecurrence[{35,-378,1264,-1272,-128},{1,41,1041,22193,433809},20] (* Harvey P. Dale, Jan 01 2019 *)
  • PARI
    Vec(x*(1 + 6*x - 16*x^2 - 8*x^3) / ((1 - 16*x)*(1 - 19*x + 74*x^2 - 80*x^3 - 8*x^4)) + O(x^30)) \\ Colin Barker, Oct 12 2017

Formula

G.f.: x*(1 +6*x -16*x^2 -8*x^3)/((1 -16*x)*(1 -19*x +74*x^2 -80*x^3 - 8*x^4)).
Showing 1-10 of 91 results. Next

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