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 11 results. Next

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)

A184048 T(n,k) = 1/9 the number of (n+1) X (k+1) 0..2 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

9, 15, 15, 25, 21, 25, 45, 31, 31, 45, 81, 51, 41, 51, 81, 153, 87, 61, 61, 87, 153, 289, 159, 97, 81, 97, 159, 289, 561, 295, 169, 117, 117, 169, 295, 561, 1089, 567, 305, 189, 153, 189, 305, 567, 1089, 2145, 1095, 577, 325, 225, 225, 325, 577, 1095, 2145, 4225, 2151
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Table starts
....9...15...25...45...81..153..289..561.1089.2145.4225..8385.16641.33153.66049
...15...21...31...51...87..159..295..567.1095.2151.4231..8391.16647.33159.66055
...25...31...41...61...97..169..305..577.1105.2161.4241..8401.16657.33169.66065
...45...51...61...81..117..189..325..597.1125.2181.4261..8421.16677.33189.66085
...81...87...97..117..153..225..361..633.1161.2217.4297..8457.16713.33225.66121
..153..159..169..189..225..297..433..705.1233.2289.4369..8529.16785.33297.66193
..289..295..305..325..361..433..569..841.1369.2425.4505..8665.16921.33433.66329
..561..567..577..597..633..705..841.1113.1641.2697.4777..8937.17193.33705.66601
.1089.1095.1105.1125.1161.1233.1369.1641.2169.3225.5305..9465.17721.34233.67129
.2145.2151.2161.2181.2217.2289.2425.2697.3225.4281.6361.10521.18777.35289.68185

Examples

			Some solutions for 6X5
..0..1..0..1..0....1..2..1..2..1....1..2..1..2..1....2..1..2..2..2
..0..1..0..1..0....1..2..1..2..1....1..0..1..0..1....0..2..0..1..0
..1..0..1..0..1....2..1..2..1..2....1..2..1..2..1....2..1..2..2..2
..0..1..0..1..0....1..2..1..2..1....1..0..1..0..1....0..2..0..1..0
..0..1..0..1..0....1..2..1..2..1....2..1..2..1..2....2..1..2..2..2
..1..0..1..0..1....2..1..2..1..2....0..1..0..1..0....0..2..0..1..0

Links

Crossrefs

Columns 1..8 are A183978(n+2), A184041, A184042, A184043, A184044, A184045, A184046, A184047.
Main diagonal is A184040.
Cf. A183986, A184039.

Programs

  • PARI
    T(n,k) = my(m=3, 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) = -7 + 4*(2^(n-1) + 2^(k-1)) + 2*(2^(floor((n-1)/2)) + 2^(floor(n/2)) + 2^(floor((k-1)/2)) + 2^(floor(k/2))). (End)

A184030 1/16 the number of (n+1) X (n+1) 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

16, 40, 82, 166, 322, 634, 1234, 2434, 4786, 9490, 18802, 37426, 74482, 148594, 296434, 592114, 1182706, 2363890, 4724722, 9446386, 18886642, 37767154, 75522034, 151031794, 302039026, 604053490, 1208057842, 2416066546, 4832034802, 9663971314, 19327746034, 38655295474
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Examples

			Some solutions for 4X4
..1..3..1..1....3..0..3..0....2..2..2..2....0..0..0..0....1..0..0..0
..1..1..1..3....0..1..0..1....1..3..1..3....2..2..2..2....0..2..1..2
..1..3..1..1....0..3..0..3....2..2..2..2....0..0..0..0....1..0..0..0
..1..1..1..3....1..0..1..0....3..1..3..1....2..2..2..2....0..2..1..2

Links

Crossrefs

Diagonal of A184039.

Programs

  • PARI
    Vec(2*(8 - 4*x - 19*x^2 + 8*x^3)/((1 - x)*(1 - 2*x)*(1 - 2*x^2)) + O(x^30)) \\ 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.: 2*x*(8 - 4*x - 19*x^2 + 8*x^3)/((1 - x)*(1 - 2*x)*(1 - 2*x^2)). (End)

Extensions

a(14) onwards from Andrew Howroyd, Mar 09 2024

A184031 1/16 the number of (n+1) X 2 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

16, 28, 49, 91, 169, 325, 625, 1225, 2401, 4753, 9409, 18721, 37249, 74305, 148225, 296065, 591361, 1181953, 2362369, 4723201, 9443329, 18883585, 37761025, 75515905, 151019521, 302026753, 604028929, 1208033281, 2416017409, 4831985665
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 1 of A184039.

Examples

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

Links

Crossrefs

Cf. A184039.

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*(16 - 20*x - 35*x^2 + 40*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 9*2^(n/2-1) + 9*2^(n-1) + 1 for n even.
a(n) = 9*2^(n-1) + 3*2^((n+1)/2) + 1 for n odd.
(End)

A184032 1/16 the number of (n+1) X 3 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

28, 40, 61, 103, 181, 337, 637, 1237, 2413, 4765, 9421, 18733, 37261, 74317, 148237, 296077, 591373, 1181965, 2362381, 4723213, 9443341, 18883597, 37761037, 75515917, 151019533, 302026765, 604028941, 1208033293, 2416017421, 4831985677
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 2 of A184039.

Examples

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

Links

Crossrefs

Cf. A184039.

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*(28 - 44*x - 59*x^2 + 88*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 9*2^(n/2-1) + 9*2^(n-1) + 13 for n even.
a(n) = 9*2^(n-1) + 3*2^((n+1)/2) + 13 for n odd.
(End)

A184033 1/16 the number of (n+1) X 4 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

49, 61, 82, 124, 202, 358, 658, 1258, 2434, 4786, 9442, 18754, 37282, 74338, 148258, 296098, 591394, 1181986, 2362402, 4723234, 9443362, 18883618, 37761058, 75515938, 151019554, 302026786, 604028962, 1208033314, 2416017442, 4831985698
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 3 of A184039.

Examples

			Some solutions for 3 X 4:
..0..0..1..2....0..1..0..1....0..2..3..2....2..2..2..2....1..3..1..3
..1..2..0..0....3..2..3..2....3..0..0..0....0..0..0..0....3..2..3..2
..0..0..1..2....1..0..1..0....0..2..3..2....2..2..2..2....1..3..1..3

Links

Crossrefs

Cf. A184039.

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*(49 - 86*x - 101*x^2 + 172*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 9*2^(n/2-1) + 9*2^(n-1) + 34 for n even.
a(n) = 9*2^(n-1) + 3*2^((n+1)/2) + 34 for n odd.
(End)

A184034 1/16 the number of (n+1) X 5 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

91, 103, 124, 166, 244, 400, 700, 1300, 2476, 4828, 9484, 18796, 37324, 74380, 148300, 296140, 591436, 1182028, 2362444, 4723276, 9443404, 18883660, 37761100, 75515980, 151019596, 302026828, 604029004, 1208033356, 2416017484, 4831985740
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 4 of A184039.

Examples

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

Links

Crossrefs

Cf. A184039.

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*(91 - 170*x - 185*x^2 + 340*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 9*2^(n/2-1) + 9*2^(n-1) + 76 for n even.
a(n) = 9*2^(n-1) + 3*2^((n+1)/2) + 76 for n odd.
(End)

A184035 1/16 the number of (n+1) X 6 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

169, 181, 202, 244, 322, 478, 778, 1378, 2554, 4906, 9562, 18874, 37402, 74458, 148378, 296218, 591514, 1182106, 2362522, 4723354, 9443482, 18883738, 37761178, 75516058, 151019674, 302026906, 604029082, 1208033434, 2416017562, 4831985818
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 5 of A184039.

Examples

			Some solutions for 3 X 6:
..1..2..0..3..0..2....0..0..3..1..3..1....3..2..2..2..3..2....0..0..2..0..0..3
..0..3..1..2..1..3....3..1..0..0..0..0....2..2..3..2..2..2....2..3..0..3..2..0
..1..2..0..3..0..2....0..0..3..1..3..1....3..2..2..2..3..2....0..0..2..0..0..3

Links

Crossrefs

Cf. A184039.

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*(169 - 326*x - 341*x^2 + 652*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 9*2^(n/2-1) + 9*2^(n-1) + 154 for n even.
a(n) = 9*2^(n-1) + 3*2^((n+1)/2) + 154 for n odd.
(End)

A184036 1/16 the number of (n+1) X 7 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

325, 337, 358, 400, 478, 634, 934, 1534, 2710, 5062, 9718, 19030, 37558, 74614, 148534, 296374, 591670, 1182262, 2362678, 4723510, 9443638, 18883894, 37761334, 75516214, 151019830, 302027062, 604029238, 1208033590, 2416017718, 4831985974
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 6 of A184039.

Examples

			Some solutions for 3 X 7:
..1..2..2..2..2..3..1....3..3..3..1..3..1..0....2..2..2..1..3..1..3
..2..3..1..3..1..2..2....0..1..0..3..0..3..3....3..1..3..2..2..2..2
..1..2..2..2..2..3..1....3..3..3..1..3..1..0....2..2..2..1..3..1..3

Links

Crossrefs

Cf. A184039.

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*(325 - 638*x - 653*x^2 + 1276*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 9*2^(n/2-1) + 9*2^(n-1) + 310 for n even.
a(n) = 9*2^(n-1) + 3*2^((n+1)/2) + 310 for n odd.
(End)

A184037 1/16 the number of (n+1) X 8 0..3 arrays with all 2 X 2 subblocks having the same four values.

Original entry on oeis.org

625, 637, 658, 700, 778, 934, 1234, 1834, 3010, 5362, 10018, 19330, 37858, 74914, 148834, 296674, 591970, 1182562, 2362978, 4723810, 9443938, 18884194, 37761634, 75516514, 151020130, 302027362, 604029538, 1208033890, 2416018018, 4831986274
Offset: 1

Views

Author

R. H. Hardin, Jan 08 2011

Keywords

Comments

Column 7 of A184039.

Examples

			Some solutions for 3 X 8:
..2..2..2..2..2..2..2..1....0..0..3..2..0..2..0..2....2..3..2..3..1..3..2..3
..3..1..3..1..3..1..3..2....3..2..0..0..3..0..3..0....1..2..1..2..2..2..1..2
..2..2..2..2..2..2..2..1....0..0..3..2..0..2..0..2....2..3..2..3..1..3..2..3

Links

Crossrefs

Cf. A184039.

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*(625 - 1238*x - 1253*x^2 + 2476*x^3) / ((1 - x)*(1 - 2*x)*(1 - 2*x^2)).
a(n) = 9*2^(n/2-1) + 9*2^(n-1) + 610 for n even.
a(n) = 9*2^(n-1) + 3*2^((n+1)/2) + 610 for n odd.
(End)
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