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 13 results. Next

A194772 Number of lower triangles of an (n+2) X (n+2) 0..2 array with new values introduced in row major order 0..2 and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

7, 17, 41, 95, 219, 493, 1101, 2427, 5311, 11529, 24881, 53399, 114083, 242725, 514581, 1087411, 2291335, 4815681, 10097401, 21126863, 44117867, 91963997, 191384541, 397682155, 825190479, 1710033273, 3539371201, 7317351687
Offset: 1

Views

Author

R. H. Hardin, Sep 02 2011

Keywords

Comments

Column 2 of A194778.

Examples

			All solutions for 3 X 3 including a 2.
..0
..1.1
..1.1.2

Links

Crossrefs

Cf. A194778.

Formula

Empirical: a(n) = 3*a(n-1) +a(n-2) -7*a(n-3) +4*a(n-5).
Conjectures from Colin Barker, May 05 2018: (Start)
G.f.: x*(7 - 4*x - 17*x^2 + 4*x^3 + 12*x^4) / ((1 - x)*(1 + x)^2*(1 - 2*x)^2).
a(n) = (-3*n + 2^(n+3)*(3*n+8) + 17) / 27 for n even.
a(n) = (3*n + 2^(n+3)*(3*n+8) + 10) / 27 for n odd.
(End)

A194773 Number of lower triangles of an (n+4) X (n+4) 0..3 array with new values introduced in row major order 0..3 and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

43, 105, 265, 655, 1641, 4069, 10121, 25025, 61809, 152019, 372979, 912017, 2224175, 5408825, 13120905, 31752431, 76673045, 184762437, 444388957, 1066954113, 2557518625, 6121119363, 14629468735, 34918297649, 83242029523, 198212839881
Offset: 1

Views

Author

R. H. Hardin, Sep 02 2011

Keywords

Comments

Column 3 of A194778.

Examples

			All solutions for 5 X 5 including a 3:
  0           0
  1 1         1 1
  1 1 2       1 1 1
  1 1 2 2     2 2 2 2
  1 1 2 2 3   2 2 2 2 3

Links

Crossrefs

Cf. A194778.

Formula

Empirical: a(n) = 5*a(n-1) -2*a(n-2) -22*a(n-3) +21*a(n-4) +35*a(n-5) -34*a(n-6) -26*a(n-7) +15*a(n-8) +9*a(n-9).
Empirical g.f.: x*(43 - 110*x - 174*x^2 + 486*x^3 + 303*x^4 - 706*x^5 - 310*x^6 + 318*x^7 + 153*x^8) / ((1 - x)*(1 - x - x^2)^2*(1 - x - 3*x^2)^2). - Colin Barker, May 05 2018

A194774 Number of lower triangles of an (n+6)X(n+6) 0..4 array with new values introduced in row major order 0..4 and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

268, 672, 1736, 4464, 11686, 30608, 80961, 214469, 570921, 1521287, 4061881, 10847647, 28982750, 77406210, 206655490, 551301206, 1469565060, 3913609398, 10412354243, 27674510615, 73480819187, 194908131709, 516485443463
Offset: 1

Views

Author

R. H. Hardin Sep 02 2011

Keywords

Comments

Column 4 of A194778

Examples

			All solutions for 7X7 including a 4
..0..............0..............0
..1.1............1.1............1.1
..1.1.2..........1.1.1..........1.1.1
..1.1.2.2........2.2.2.2........2.2.2.2
..1.1.2.2.3......2.2.2.2.2......2.2.2.2.3
..1.1.2.2.3.3....3.3.3.3.3.3....2.2.2.2.3.3
..1.1.2.2.3.3.4..3.3.3.3.3.3.4..2.2.2.2.3.3.4

Links

A194775 Number of lower triangles of an (n+8) X (n+8) 0..5 array with new values introduced in row major order 0..5 and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1740, 4490, 11857, 31429, 84727, 229869, 631373, 1745459, 4867831, 13649561, 38503606, 109050858, 310061698, 883972100, 2526165997, 7230962925, 20725462609, 59454601239, 170659921683, 490024454069, 1407237659011
Offset: 1

Views

Author

R. H. Hardin, Sep 02 2011

Keywords

Comments

Column 5 of A194778.

Examples

			All solutions for 9 X 9 including a 5:
  0                  0                  0                  0
  1 1                1 1                1 1                1 1
  1 1 2              1 1 1              1 1 1              1 1 1
  1 1 2 2            2 2 2 2            2 2 2 2            2 2 2 2
  1 1 2 2 3          2 2 2 2 3          2 2 2 2 2          2 2 2 2 2
  1 1 2 2 3 3        2 2 2 2 3 3        3 3 3 3 3 3        3 3 3 3 3 3
  1 1 2 2 3 3 4      2 2 2 2 3 3 4      3 3 3 3 3 3 3      3 3 3 3 3 3 4
  1 1 2 2 3 3 4 4    2 2 2 2 3 3 4 4    4 4 4 4 4 4 4 4    3 3 3 3 3 3 4 4
  1 1 2 2 3 3 4 4 5  2 2 2 2 3 3 4 4 5  4 4 4 4 4 4 4 4 5  3 3 3 3 3 3 4 4 5

A194776 Number of lower triangles of an (n+10) X (n+10) 0..6 array with new values introduced in row major order 0..6 and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

11862, 31466, 85007, 231395, 639325, 1782111, 5029741, 14321783, 41203606, 119499674, 349498634, 1029170664, 3050715488, 9092629968, 27237137985, 81930133845, 247359353107, 749090477957
Offset: 1

Views

Author

R. H. Hardin, Sep 02 2011

Keywords

Comments

Column 6 of A194778.

Examples

			Some solutions for 11 X 11 including a 6:
  0                      0                      0
  1 1                    1 1                    1 1
  1 1 2                  1 1 1                  1 1 1
  1 1 2 2                2 2 2 2                2 2 2 2
  1 1 2 2 3              2 2 2 2 3              2 2 2 2 2
  1 1 2 2 3 3            2 2 2 2 3 3            3 3 3 3 3 3
  1 1 2 2 3 3 4          2 2 2 2 3 3 4          3 3 3 3 3 3 4
  1 1 2 2 3 3 4 4        2 2 2 2 3 3 4 4        3 3 3 3 3 3 4 4
  1 1 2 2 3 3 4 4 5      2 2 2 2 3 3 4 4 5      3 3 3 3 3 3 4 4 5
  1 1 2 2 3 3 4 4 5 5    2 2 2 2 3 3 4 4 5 5    3 3 3 3 3 3 4 4 5 5
  1 1 2 2 3 3 4 4 5 5 6  2 2 2 2 3 3 4 4 5 5 6  3 3 3 3 3 3 4 4 5 5 6

A194777 Number of lower triangles of an (n+12) X (n+12) 0..7 array with new values introduced in row major order 0..7 and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

85013, 231445, 639753, 1784723, 5044981, 14399939, 41587186, 121261490, 357312638, 1062459348, 3188769448, 9649809740, 29439886715, 90469050263, 279932030931, 871510145365
Offset: 1

Views

Author

R. H. Hardin, Sep 02 2011

Keywords

Comments

Column 7 of A194778.

Examples

			Some solutions for 13 X 13 including a 7:
  0                           0
  1 1                         1 1
  1 1 1                       1 1 1
  2 2 2 2                     2 2 2 2
  2 2 2 2 2                   2 2 2 2 2
  3 3 3 3 3 3                 3 3 3 3 3 3
  3 3 3 3 3 3 3               3 3 3 3 3 3 3
  4 4 4 4 4 4 4 4             4 4 4 4 4 4 4 4
  4 4 4 4 4 4 4 4 4           4 4 4 4 4 4 4 4 4
  5 5 5 5 5 5 5 5 5 5         5 5 5 5 5 5 5 5 5 5
  5 5 5 5 5 5 5 5 5 5 6       5 5 5 5 5 5 5 5 5 5 5
  5 5 5 5 5 5 5 5 5 5 6 6     6 6 6 6 6 6 6 6 6 6 6 6
  5 5 5 5 5 5 5 5 5 5 6 6 7   6 6 6 6 6 6 6 6 6 6 6 6 7

A194779 Number of lower triangles of a (2n-1)X(2n-1) 0..n array with new values introduced in row major order 0..n and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

1, 7, 43, 268, 1740, 11862, 85013, 639760, 5045610, 41615165, 358185399, 3210667490, 29914284833, 289187621900, 2895861006415
Offset: 1

Views

Author

R. H. Hardin Sep 02 2011

Keywords

Comments

Row 1 of A194778

Examples

			All solutions for n=2 containing a 2
..0
..1.1
..1.1.2

A194780 Number of lower triangles of a (2n)X(2n) 0..n array with new values introduced in row major order 0..n and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

3, 17, 105, 672, 4490, 31466, 231445, 1784788, 14404218, 121421075, 1066906979, 9753108510, 92586250999, 911156724048
Offset: 1

Views

Author

R. H. Hardin Sep 02 2011

Keywords

Comments

Row 2 of A194778

Examples

			All solutions for n=2 containing a 2
..0........0........0........0........0
..0.0......1.1......1.1......0.1......1.1
..1.1.1....1.1.1....1.1.2....0.1.1....1.1.1
..1.1.1.2..1.1.1.2..1.1.2.2..0.1.1.2..2.2.2.2

A194781 Number of lower triangles of a (2n+1)X(2n+1) 0..n array with new values introduced in row major order 0..n and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

6, 41, 265, 1736, 11857, 85007, 639753, 5045602, 41615156, 358185389, 3210667479, 29914284821, 289187621887
Offset: 1

Views

Author

R. H. Hardin Sep 02 2011

Keywords

Comments

Row 3 of A194778

Examples

			Some solutions for n=2 containing a 2
..0..........0..........0..........0..........0..........0..........0
..1.1........1.1........1.1........0.0........1.1........1.1........0.1
..1.1.2......1.1.1......1.1.1......1.1.1......1.1.1......1.1.2......0.1.1
..1.1.2.2....1.1.1.1....1.1.1.2....1.1.1.1....2.2.2.2....1.1.2.2....0.1.1.1
..1.1.2.2.1..2.2.2.2.2..1.1.1.2.2..1.1.1.1.2..2.2.2.2.0..1.1.2.2.0..0.1.1.1.2

A194782 Number of lower triangles of a (2n+2)X(2n+2) 0..n array with new values introduced in row major order 0..n and no element unequal to more than one horizontal or vertical neighbor.

Original entry on oeis.org

12, 95, 655, 4464, 31429, 231395, 1784723, 14404136, 121420974, 1066906857, 9753108365, 92586250829, 911156723851
Offset: 1

Views

Author

R. H. Hardin Sep 02 2011

Keywords

Comments

Row 4 of A194778

Examples

			Some solutions for n=2 containing a 2
..0............0............0............0............0............0
..1.1..........0.0..........0.0..........0.0..........0.0..........1.1
..1.1.1........1.1.1........1.1.1........0.0.0........0.0.0........1.1.1
..0.0.0.0......1.1.1.1......1.1.1.1......0.0.0.1......1.1.1.1......1.1.1.2
..0.0.0.0.0....1.1.1.1.1....1.1.1.1.1....0.0.0.1.1....1.1.1.1.1....1.1.1.2.2
..2.2.2.2.2.2..2.2.2.2.2.2..1.1.1.1.1.2..0.0.0.1.1.2..1.1.1.1.1.2..1.1.1.2.2.0
Showing 1-10 of 13 results. Next

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