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 10 results.

A183986 T(n,k) = 1/4 the number of (n+1) X (k+1) binary arrays with all 2 X 2 subblock sums the same.

Original entry on oeis.org

4, 6, 6, 9, 8, 9, 15, 11, 11, 15, 25, 17, 14, 17, 25, 45, 27, 20, 20, 27, 45, 81, 47, 30, 26, 30, 47, 81, 153, 83, 50, 36, 36, 50, 83, 153, 289, 155, 86, 56, 46, 56, 86, 155, 289, 561, 291, 158, 92, 66, 66, 92, 158, 291, 561, 1089, 563, 294, 164, 102, 86, 102, 164, 294, 563
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Table starts
...4...6...9..15..25..45..81.153.289..561.1089.2145.4225.8385.16641.33153.66049
...6...8..11..17..27..47..83.155.291..563.1091.2147.4227.8387.16643.33155.66051
...9..11..14..20..30..50..86.158.294..566.1094.2150.4230.8390.16646.33158.66054
..15..17..20..26..36..56..92.164.300..572.1100.2156.4236.8396.16652.33164.66060
..25..27..30..36..46..66.102.174.310..582.1110.2166.4246.8406.16662.33174.66070
..45..47..50..56..66..86.122.194.330..602.1130.2186.4266.8426.16682.33194.66090
..81..83..86..92.102.122.158.230.366..638.1166.2222.4302.8462.16718.33230.66126
.153.155.158.164.174.194.230.302.438..710.1238.2294.4374.8534.16790.33302.66198
.289.291.294.300.310.330.366.438.574..846.1374.2430.4510.8670.16926.33438.66334
.561.563.566.572.582.602.638.710.846.1118.1646.2702.4782.8942.17198.33710.66606

Examples

			Some solutions for 6 X 5
..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
..1..0..1..1..0....1..0..1..0..1....0..0..1..1..0....0..0..1..0..1
..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
..1..0..1..1..0....0..1..0..1..0....0..0..1..1..0....0..0..1..0..1
..0..1..0..0..1....1..1..1..1..1....1..1..0..0..1....1..1..0..1..0
..1..0..1..1..0....0..1..0..1..0....0..0..1..1..0....0..0..1..0..1

Links

Crossrefs

Columns 1..8 are A183978, A183979, A183980, A183981, A183982, A183983, A183984, A183985.
Main diagonal is A183977.
Cf. A184039, A184048.

Programs

  • PARI
    T(n,k) = my(m=2, b=t->2^t-1); m^2 + (m-1)^2*(b(n-1) + b(k-1)) + (m-1)*(b((n-1)\2) + b(n\2) + b((k-1)\2) + b(k\2)) \\ Andrew Howroyd, Mar 09 2024

Formula

Empirical, for every row and column: a(n) = 3*a(n-1)-6*a(n-3)+4*a(n-4).
From Andrew Howroyd, Mar 09 2024: (Start)
The above empirical formula is correct.
T(n,k) = -2 + 2^(n-1) + 2^(k-1) + 2^(floor((n-1)/2)) + 2^(floor(n/2)) + 2^(floor((k-1)/2)) + 2^(floor(k/2)). (End)

A184039 T(n,k) = 1/16 the number of (n+1) X (k+1) 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

16, 28, 28, 49, 40, 49, 91, 61, 61, 91, 169, 103, 82, 103, 169, 325, 181, 124, 124, 181, 325, 625, 337, 202, 166, 202, 337, 625, 1225, 637, 358, 244, 244, 358, 637, 1225, 2401, 1237, 658, 400, 322, 400, 658, 1237, 2401, 4753, 2413, 1258, 700, 478, 478, 700, 1258
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Table starts
...16...28...49...91..169..325..625.1225.2401.4753..9409.18721.37249.74305
...28...40...61..103..181..337..637.1237.2413.4765..9421.18733.37261.74317
...49...61...82..124..202..358..658.1258.2434.4786..9442.18754.37282.74338
...91..103..124..166..244..400..700.1300.2476.4828..9484.18796.37324.74380
..169..181..202..244..322..478..778.1378.2554.4906..9562.18874.37402.74458
..325..337..358..400..478..634..934.1534.2710.5062..9718.19030.37558.74614
..625..637..658..700..778..934.1234.1834.3010.5362.10018.19330.37858.74914
.1225.1237.1258.1300.1378.1534.1834.2434.3610.5962.10618.19930.38458.75514
.2401.2413.2434.2476.2554.2710.3010.3610.4786.7138.11794.21106.39634.76690
.4753.4765.4786.4828.4906.5062.5362.5962.7138.9490.14146.23458.41986.79042

Examples

			Some solutions for 4X3
..0..3..0....3..2..3....3..2..3....1..0..1....2..3..2....3..2..2....3..1..3
..3..2..3....3..3..3....1..3..1....2..1..2....3..2..3....2..1..3....1..2..1
..3..0..3....3..2..3....3..2..3....0..1..0....2..3..2....3..2..2....3..1..3
..2..3..2....3..3..3....1..3..1....1..2..1....2..3..2....2..1..3....2..1..2

Links

Crossrefs

Columns 1..8 are A184031, A184032, A184033, A184034, A184035, A184036, A184037, A184038.
Main diagonal is A184030.
Cf. A183986, A184048.

Programs

  • PARI
    T(n,k) = my(m=4, b=t->2^t-1); m^2 + (m-1)^2*(b(n-1) + b(k-1)) + (m-1)*(b((n-1)\2) + b(n\2) + b((k-1)\2) + b(k\2)) \\ Andrew Howroyd, Mar 09 2024

Formula

Empirical, for all rows and columns: a(n)=3*a(n-1)-6*a(n-3)+4*a(n-4).
From Andrew Howroyd, Mar 09 2024: (Start)
The above empirical formula is correct.
T(n,k) = -14 + 9*(2^(n-1) + 2^(k-1)) + 3*(2^(floor((n-1)/2)) + 2^(floor(n/2)) + 2^(floor((k-1)/2)) + 2^(floor(k/2))). (End)

A184040 1/9 the number of (n+1) X (n+1) 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

9, 21, 41, 81, 153, 297, 569, 1113, 2169, 4281, 8441, 16761, 33273, 66297, 132089, 263673, 526329, 1051641, 2101241, 4200441, 8396793, 16789497, 33570809, 67133433, 134250489, 268484601, 536936441, 1073840121, 2147614713, 4295163897, 8590196729, 17180262393
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Examples

			Some solutions for 5X5
..1..0..1..0..1....1..2..1..0..1....1..0..1..0..1....0..1..0..1..0
..1..0..1..0..1....0..0..0..2..0....2..0..2..0..2....1..1..1..1..1
..0..1..0..1..0....1..2..1..0..1....0..1..0..1..0....1..0..1..0..1
..0..1..0..1..0....0..0..0..2..0....0..2..0..2..0....1..1..1..1..1
..0..1..0..1..0....1..2..1..0..1....1..0..1..0..1....0..1..0..1..0

Links

Crossrefs

Diagonal of A184048.

Programs

  • PARI
    Vec((9 - 6*x - 22*x^2 + 12*x^3)/((1 - x)*(1 - 2*x)*(1 - 2*x^2)) + O(x^32)) \\ Andrew Howroyd, Mar 09 2024

Formula

From Andrew Howroyd, Mar 09 2024: (Start)
a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
G.f.: x*(9 - 6*x - 22*x^2 + 12*x^3)/((1 - x)*(1 - 2*x)*(1 - 2*x^2)). (End)

Extensions

a(15) onwards from Andrew Howroyd, Mar 09 2024

A184041 1/9 the number of (n+1) X 3 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

15, 21, 31, 51, 87, 159, 295, 567, 1095, 2151, 4231, 8391, 16647, 33159, 66055, 131847, 263175, 525831, 1050631, 2100231, 4198407, 8394759, 16785415, 33566727, 67125255, 134242311, 268468231, 536920071, 1073807367, 2147581959, 4295098375
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 2 of A184048.

Examples

			Some solutions for 5 X 3:
..1..0..0....2..1..2....2..1..2....0..1..0....0..0..0....0..1..0....1..1..1
..0..1..1....0..0..0....1..1..1....0..2..0....1..1..1....1..2..1....0..1..0
..1..0..0....2..1..2....2..1..2....1..0..1....0..0..0....1..0..1....1..1..1
..0..1..1....0..0..0....1..1..1....0..2..0....1..1..1....2..1..2....0..1..0
..1..0..0....1..2..1....1..2..1....1..0..1....0..0..0....0..1..0....1..1..1

Links

Crossrefs

Cf. A184048.

Formula

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
Conjectures from Colin Barker, Apr 10 2018: (Start)
G.f.: x*(15 - 24*x - 32*x^2 + 48*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2) + 2^(n+1) + 7 for n even.
a(n) = 2^(n+1) + 2^((n+3)/2) + 7 for n odd.
(End)

A184042 1/9 the number of (n+1) X 4 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

25, 31, 41, 61, 97, 169, 305, 577, 1105, 2161, 4241, 8401, 16657, 33169, 66065, 131857, 263185, 525841, 1050641, 2100241, 4198417, 8394769, 16785425, 33566737, 67125265, 134242321, 268468241, 536920081, 1073807377, 2147581969, 4295098385
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 3 of A184048.

Examples

			Some solutions for 5 X 4:
..2..1..2..1....0..2..0..2....2..2..2..0....1..1..1..1....0..2..0..2
..0..2..0..2....1..1..1..1....0..0..0..2....2..1..2..1....2..0..2..0
..1..2..1..2....2..0..2..0....2..2..2..0....1..1..1..1....0..2..0..2
..0..2..0..2....1..1..1..1....0..0..0..2....2..1..2..1....2..0..2..0
..1..2..1..2....2..0..2..0....2..2..2..0....1..1..1..1....2..0..2..0

Links

Crossrefs

Cf. A184048.

Formula

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
Conjectures from Colin Barker, Apr 10 2018: (Start)
G.f.: x*(25 - 44*x - 52*x^2 + 88*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2) + 2^(n+1) + 17 for n even.
a(n) = 2^(n+1) + 2^((n+3)/2) + 17 for n odd.
(End)

A184043 1/9 the number of (n+1) X 5 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

45, 51, 61, 81, 117, 189, 325, 597, 1125, 2181, 4261, 8421, 16677, 33189, 66085, 131877, 263205, 525861, 1050661, 2100261, 4198437, 8394789, 16785445, 33566757, 67125285, 134242341, 268468261, 536920101, 1073807397, 2147581989, 4295098405
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 4 of A184048.

Examples

			Some solutions for 7 X 5:
..0..1..0..1..0....2..0..2..0..2....1..2..1..2..1....0..0..0..1..0
..0..1..0..1..0....2..2..2..2..2....0..0..0..0..0....1..1..1..0..1
..1..0..1..0..1....0..2..0..2..0....1..2..1..2..1....0..0..0..1..0
..0..1..0..1..0....2..2..2..2..2....0..0..0..0..0....1..1..1..0..1
..0..1..0..1..0....0..2..0..2..0....2..1..2..1..2....0..0..0..1..0
..0..1..0..1..0....2..2..2..2..2....0..0..0..0..0....1..1..1..0..1
..1..0..1..0..1....2..0..2..0..2....2..1..2..1..2....0..0..0..1..0

Links

Crossrefs

Cf. A184048.

Formula

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
Conjectures from Colin Barker, Apr 10 2018: (Start)
G.f.: x*(45 - 84*x - 92*x^2 + 168*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2) + 2^(n+1) + 37 for n even.
a(n) = 2^(n+1) + 2^((n+3)/2) + 37 for n odd.
(End)

A184044 1/9 the number of (n+1) X 6 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

81, 87, 97, 117, 153, 225, 361, 633, 1161, 2217, 4297, 8457, 16713, 33225, 66121, 131913, 263241, 525897, 1050697, 2100297, 4198473, 8394825, 16785481, 33566793, 67125321, 134242377, 268468297, 536920137, 1073807433, 2147582025, 4295098441, 8590131273, 17180131401
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Examples

			Some solutions for 5 X 6:
..2..0..1..0..2..1....2..1..2..0..2..0....0..2..1..0..1..0....1..2..1..0..1..0
..1..1..2..1..1..0....2..0..2..1..2..1....1..0..0..2..0..2....2..0..2..2..2..2
..2..0..1..0..2..1....2..1..2..0..2..0....0..2..1..0..1..0....1..2..1..0..1..0
..1..1..2..1..1..0....2..0..2..1..2..1....1..0..0..2..0..2....2..0..2..2..2..2
..2..0..1..0..2..1....2..1..2..0..2..0....0..2..1..0..1..0....1..2..1..0..1..0

Links

Crossrefs

Column 5 of A184048.

Formula

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
Conjectures from Colin Barker, Apr 10 2018: (Start)
G.f.: x*(81 - 156*x - 164*x^2 + 312*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2) + 2^(n+1) + 73 for n even.
a(n) = 2^(n+1) + 2^((n+3)/2) + 73 for n odd. (End)
Conjectured e.g.f.: 2*cosh(2*x) + 3*cosh(sqrt(2)*x) + 73*sinh(x) + cosh(x)*(73 + 4*sinh(x)) + 2*sqrt(2)*sinh(sqrt(2)*x) - 78. - Stefano Spezia, Aug 01 2025

A184045 1/9 the number of (n+1) X 7 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

153, 159, 169, 189, 225, 297, 433, 705, 1233, 2289, 4369, 8529, 16785, 33297, 66193, 131985, 263313, 525969, 1050769, 2100369, 4198545, 8394897, 16785553, 33566865, 67125393, 134242449, 268468369, 536920209, 1073807505, 2147582097, 4295098513
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 6 of A184048.

Examples

			Some solutions for 5 X 7:
..2..1..0..1..2..1..2....2..1..2..1..2..1..2....1..2..2..0..2..0..2
..0..2..2..2..0..2..0....2..1..2..1..2..1..2....2..0..1..2..1..2..1
..2..1..0..1..2..1..2....1..2..1..2..1..2..1....1..2..2..0..2..0..2
..0..2..2..2..0..2..0....2..1..2..1..2..1..2....2..0..1..2..1..2..1
..2..1..0..1..2..1..2....2..1..2..1..2..1..2....1..2..2..0..2..0..2

Links

Crossrefs

Cf. A184048.

Formula

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
Conjectures from Colin Barker, Apr 10 2018: (Start)
G.f.: x*(153 - 300*x - 308*x^2 + 600*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2) + 2^(n+1) + 145 for n even.
a(n) = 2^(n+1) + 2^((n+3)/2) + 145 for n odd.
(End)

A184046 1/9 the number of (n+1) X 8 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

289, 295, 305, 325, 361, 433, 569, 841, 1369, 2425, 4505, 8665, 16921, 33433, 66329, 132121, 263449, 526105, 1050905, 2100505, 4198681, 8395033, 16785689, 33567001, 67125529, 134242585, 268468505, 536920345, 1073807641, 2147582233, 4295098649
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 7 of A184048.

Examples

			Some solutions for 5 X 8:
..1..1..1..0..0..0..1..1....0..2..0..2..2..2..0..2....1..0..1..0..1..0..1..0
..0..0..0..1..1..1..0..0....2..1..2..1..0..1..2..1....2..1..2..1..2..1..2..1
..1..1..1..0..0..0..1..1....0..2..0..2..2..2..0..2....1..0..1..0..1..0..1..0
..0..0..0..1..1..1..0..0....2..1..2..1..0..1..2..1....1..2..1..2..1..2..1..2
..1..1..1..0..0..0..1..1....0..2..0..2..2..2..0..2....0..1..0..1..0..1..0..1

Links

Crossrefs

Cf. A184048.

Formula

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
Conjectures from Colin Barker, Apr 10 2018: (Start)
G.f.: x*(289 - 572*x - 580*x^2 + 1144*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2) + 2^(n+1) + 281 for n even.
a(n) = 2^(n+1) + 2^((n+3)/2) + 281 for n odd.
(End)

A184047 1/9 the number of (n+1) X 9 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

561, 567, 577, 597, 633, 705, 841, 1113, 1641, 2697, 4777, 8937, 17193, 33705, 66601, 132393, 263721, 526377, 1051177, 2100777, 4198953, 8395305, 16785961, 33567273, 67125801, 134242857, 268468777, 536920617, 1073807913, 2147582505, 4295098921
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 8 of A184048.

Examples

			Some solutions for 5 X 9:
..2..0..2..0..0..1..0..1..2....1..1..0..0..0..0..1..0..1
..0..1..0..1..2..0..2..0..0....0..0..1..1..1..1..0..1..0
..2..0..2..0..0..1..0..1..2....1..1..0..0..0..0..1..0..1
..0..1..0..1..2..0..2..0..0....0..0..1..1..1..1..0..1..0
..2..0..2..0..0..1..0..1..2....1..1..0..0..0..0..1..0..1

Links

Crossrefs

Cf. A184048.

Formula

Empirical: a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4).
Conjectures from Colin Barker, Apr 10 2018: (Start)
G.f.: x*(561 - 1116*x - 1124*x^2 + 2232*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 3*2^(n/2) + 2^(n+1) + 553 for n even.
a(n) = 2^(n+1) + 2^((n+3)/2) + 553 for n odd.
(End)
Showing 1-10 of 10 results.

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