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 1-8 of 8 results.

A259215 Number of (n+1) X (1+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

7, 13, 24, 45, 85, 162, 311, 601, 1168, 2281, 4473, 8802, 17371, 34365, 68120, 135253, 268909, 535234, 1066287, 2125809, 4240672, 8463633, 16898609, 33750850, 67426675, 134731957, 269267496, 538217181, 1075920133, 2151008226, 4300670183
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Examples

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

Links

Crossrefs

Column 1 of A259222.

Formula

Empirical: a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).
Conjectures from Colin Barker, Dec 24 2018: (Start)
G.f.: x*(7 - 8*x - 8*x^2) / ((1 - 2*x)*(1 - x - x^2)).
a(n) = 2^(1+n) + (2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5)) + (1+sqrt(5))^n*(2+sqrt(5)))) / sqrt(5).
(End)

A259216 Number of (n+1) X (2+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

13, 23, 40, 71, 127, 230, 421, 779, 1456, 2747, 5227, 10022, 19345, 37559, 73288, 143615, 282439, 557126, 1101709, 2183123, 4333408, 8613683, 17141395, 34143686, 68062297, 135760415, 270931576, 540909719, 1080276751, 2158057382, 4312075957
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Examples

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

Links

Crossrefs

Column 2 of A259222.

Formula

a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3).
From Colin Barker, Dec 24 2018: (Start)
G.f.: x*(13 - 16*x - 16*x^2) / ((1 - 2*x)*(1 - x - x^2)).
a(n) = 2^(1+n) + (3*2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5)) + (1+sqrt(5))^n*(2+sqrt(5)))) / sqrt(5).
(End)
a(n) = 2^(n+1)+3*A000045(n+3). - R. J. Mathar, Oct 09 2020

A259217 Number of (n+1) X (3+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

24, 40, 66, 112, 192, 334, 588, 1048, 1890, 3448, 6360, 11854, 22308, 42352, 81042, 156160, 302736, 589966, 1154844, 2269096, 4472514, 8838760, 17505576, 34732942, 69015732, 137303104, 273427698, 544948528, 1086811680, 2168631118, 4329184620
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Examples

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

Links

Crossrefs

Column 3 of A259222.

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4).
G.f.: 2*x*(12 - 28*x + x^2 + 16*x^3) / ((1 - x)*(1 - 2*x)*(1 - x - x^2)). - Colin Barker, Dec 24 2018
a(n) = 2^(n+1)+2+6*A000045(n+3). - R. J. Mathar, Oct 09 2020

A259218 Number of (n+1) X (4+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

45, 71, 112, 183, 303, 510, 869, 1499, 2616, 4619, 8251, 14910, 27249, 50343, 93968, 177071, 336567, 644702, 1243405, 2412387, 4704360, 9213891, 18112547, 35715038, 70604793, 139874255, 277587904, 551679879, 1097703231, 2186254014, 4357699061
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Examples

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

Links

Crossrefs

Column 4 of A259222.

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4).
G.f.: x*(45 - 109*x + 8*x^2 + 64*x^3) / ((1 - x)*(1 - 2*x)*(1 - x - x^2)). - Colin Barker, Dec 24 2018
a(n) = 2^(n+1)+8+11*A000045(n+3). - R. J. Mathar, Oct 09 2020

A259219 Number of (n+1) X (5+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

85, 127, 192, 303, 487, 798, 1325, 2227, 3784, 6499, 11283, 19806, 35161, 63135, 114656, 210535, 390703, 732286, 1385109, 2641659, 5075320, 9814107, 19083707, 37286398, 73147297, 143988103, 284244240, 562450047, 1115129719, 2214450654
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Examples

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

Links

Crossrefs

Column 5 of A259222.

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4).
G.f.: x*(85 - 213*x + 24*x^2 + 128*x^3) / ((1 - x)*(1 - 2*x)*(1 - x - x^2)). - Colin Barker, Dec 24 2018
From the above formulae, a(n) = 2^(n+1) + 19*Fibonacci(n+3) + 24. - Ehren Metcalfe, Dec 27 2018

A259220 Number of (n+1) X (6+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

162, 230, 334, 510, 798, 1278, 2078, 3422, 5694, 9566, 16222, 27774, 48030, 83934, 148286, 264926, 478686, 874622, 1615390, 3014238, 5678142, 10789470, 20661854, 39839870, 77278878, 150673118, 295060798, 579951582, 1143447774, 2260270206
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Examples

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

Links

Crossrefs

Column 6 of A259222.

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4).
G.f.: 2*x*(81 - 209*x + 31*x^2 + 128*x^3) / ((1 - x)*(1 - 2*x)*(1 - x - x^2)). - Colin Barker, Dec 24 2018
From the above formulae, a(n) = 2*(2^n + 16*Fibonacci(n+3) + 31). - Ehren Metcalfe, Dec 27 2018

A259221 Number of (n+1) X (7+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

311, 421, 588, 869, 1325, 2078, 3319, 5377, 8804, 14545, 24225, 40670, 68843, 117557, 202636, 352813, 620837, 1104574, 1987407, 3616121, 6651956, 12365081, 23211193, 43964734, 83952995, 161472013, 312533724, 608223317, 1189192349, 2334286430
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Examples

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

Links

Crossrefs

Column 7 of A259222.

Formula

a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4).
G.f.: x*(311 - 823*x + 148*x^2 + 512*x^3) / ((1 - x)*(1 - 2*x)*(1 - x - x^2)). - Colin Barker, Dec 24 2018
From the above formulae, a(n) = 2^(n+1) + 53*Fibonacci(n+3) + 148. - Ehren Metcalfe, Dec 27 2018

A259214 Number of (n+1)X(n+1) 0..1 arrays with each 2X2 subblock having clockwise pattern 0000 0011 or 0101.

Original entry on oeis.org

7, 23, 66, 183, 487, 1278, 3319, 8591, 22210, 57455, 148815, 386046, 1002991, 2609559, 6797794
Offset: 1

Views

Author

R. H. Hardin, Jun 21 2015

Keywords

Comments

Diagonal of A259222

Examples

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

Crossrefs

Cf. A259222

Formula

Empirical: a(n) = 6*a(n-1) -10*a(n-2) -2*a(n-3) +16*a(n-4) -6*a(n-5) -5*a(n-6) +2*a(n-7)
Empirical g.f: -x*(7-19*x-2*x^2+31*x^3-17*x^4-8*x^5+4*x^6)/(x-1)/(2*x-1)/(1+x)/(x^2-3*x+1)/(x^2+x-1) . - R. J. Mathar, Nov 09 2018
Empirical: a(n) = 18*A001906(n+1)/5 -7*A001906(n)/5 +2 + 4*2^n - 4*A000045(n+3)+2*(-1)^n /5. - R. J. Mathar, Nov 09 2018
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.