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 12 results. Next

A195098 Interspersion fractally induced by (1+[3n/4]), where [ ] = floor; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 14, 16, 17, 18, 21, 19, 20, 22, 23, 24, 28, 25, 26, 27, 29, 30, 31, 36, 32, 33, 34, 35, 37, 38, 39, 45, 40, 41, 42, 44, 43, 46, 47, 48, 55, 49, 50, 51, 54, 52, 53, 56, 57, 58, 66, 59, 60, 61, 65, 62, 63, 64, 67, 68, 69
Offset: 1

Views

Author

Clark Kimberling, Sep 08 2011

Keywords

Comments

See A194959 for a discussion of fractalization and the interspersion fractally induced by a sequence. Every pair of rows eventually intersperse. As a sequence, A194998 is a permutation of the positive integers, with inverse A195099.

Examples

			Northwest corner:
1...2...4...7...11..16..22
3...5...8...12..17..23..30
6...9...13..18..24..31..39
10..15..21..28..36..45..55
14..19..25..32..40..49..59

Crossrefs

Cf. A194959, A037915, A195097, A195099.

Programs

  • Mathematica
    r = 3/4; p[n_] := 1 + Floor[n*r] (* A037915 *)
Table[p[n], {n, 1, 90}]
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20]    (* A195097 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]](* A195098 *)
q[n_] := Position[w, n]; Flatten[Table[q[n],
{n, 1, 80}]](* A195099 *)

A194992 Number of lower triangles of an n X n 0..2 array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

3, 6, 20, 80, 576, 4608, 69632, 1114112, 34603008, 1107296256, 69793218560, 4466765987840, 567347999932416, 72620543991349248, 18518801667747479552, 4740813226943354765312, 2422574005712127994626048
Offset: 1

Views

Author

R. H. Hardin, Sep 07 2011

Keywords

Comments

Column 2 of A194998.

Examples

			Some solutions for 4 X 4
..2........1........0........2........0........0........0........0
..1.0......2.1......1.2......1.0......1.0......1.0......1.0......1.0
..2.1.2....1.2.1....0.1.0....2.1.2....2.1.0....0.1.0....2.1.0....2.1.0
..1.0.1.0..2.1.2.1..1.2.1.0..1.2.1.0..1.2.1.2..1.0.1.2..1.2.1.0..1.0.1.2

Links

Crossrefs

Cf. A194998.

A194993 Number of lower triangles of an n X n 0..3 array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

4, 10, 42, 248, 2290, 31042, 641376, 19753266, 918924546, 64065187024, 6724991990090, 1059952700132050, 251257954485037072, 89488725743024413554, 47916300899545924260762, 38558096411377540692559456
Offset: 1

Views

Author

R. H. Hardin Sep 07 2011

Keywords

Comments

Column 3 of A194998

Examples

			Some solutions for 4X4
..1........3........2........3........1........2........1........3
..2.3......2.1......1.2......2.1......0.1......3.2......2.1......2.3
..1.2.3....1.2.3....2.1.2....3.2.3....1.2.3....2.1.2....1.2.1....3.2.1
..0.1.2.3..0.1.2.3..3.2.3.2..2.3.2.3..2.3.2.1..3.2.1.2..2.1.2.3..2.1.0.1

Links

A194994 Number of lower triangles of an n X n 0..4 array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

5, 14, 66, 458, 4990, 81014, 2059822, 78060014, 4641074078, 410310652370, 57068977108754, 11768868532345686, 3829513291002131546, 1842092248483706146126, 1402261360946084657783246
Offset: 1

Views

Author

R. H. Hardin Sep 07 2011

Keywords

Comments

Column 4 of A194998

Examples

			Some solutions for 4X4
..3........2........2........2........2........4........1........4
..2.1......1.0......3.2......3.4......3.2......3.2......2.3......3.2
..3.2.3....0.1.2....2.3.2....2.3.4....2.3.2....2.3.4....3.2.3....4.3.4
..2.1.2.3..1.0.1.2..1.2.3.2..1.2.3.4..1.2.1.2..1.2.3.4..2.1.2.1..3.4.3.2

Links

A194995 Number of lower triangles of an n X n 0..5 array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

6, 18, 90, 672, 7858, 138956, 3816148, 159427052, 10305467946, 1016480307410, 154783529107972, 36032460970618398, 12930394865167766832, 7102572782942842968972, 6007897214058607933844528
Offset: 1

Views

Author

R. H. Hardin Sep 07 2011

Keywords

Comments

Column 5 of A194998

Examples

			Some solutions for 4X4
..1........4........2........1........1........3........2........4
..2.3......3.4......3.4......2.3......2.1......4.5......3.2......5.4
..3.2.1....4.3.4....2.3.2....3.2.3....1.2.3....5.4.3....2.3.4....4.3.2
..4.3.2.1..5.4.3.2..3.4.3.4..4.3.4.3..2.1.2.1..4.5.4.5..3.4.5.4..3.2.1.0

Links

A194996 Number of lower triangles of an n X n 0..6 array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

7, 22, 114, 888, 10804, 199988, 5740166, 251606900, 17066691742, 1770282323740, 283975401794580, 69694377176377328, 26447798453646675184, 15357128901797793996652, 13788844798416302935620998
Offset: 1

Views

Author

R. H. Hardin Sep 07 2011

Keywords

Comments

Column 6 of A194998

Examples

			Some solutions for 4X4
..5........4........2........1........5........4........3........2
..6.5......5.4......3.2......2.3......4.3......5.6......4.5......3.2
..5.4.3....4.5.4....2.1.2....3.4.3....3.4.3....4.5.6....3.4.5....4.3.4
..4.5.4.3..3.4.5.6..1.2.1.0..4.3.2.3..2.3.4.5..5.6.5.4..4.5.6.5..5.4.3.4

Links

A194997 Number of lower triangles of an n X n 0..7 array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

8, 26, 138, 1104, 13754, 261324, 7686580, 346018484, 24069199390, 2566345975600, 422456182660436, 106689940668487188, 41576384767924321080, 24871115264481653525780, 22948996533081051784100632
Offset: 1

Views

Author

R. H. Hardin Sep 07 2011

Keywords

Comments

Column 7 of A194998

Examples

			Some solutions for 4X4
..2........6........2........2........2........0........2........2
..3.2......7.6......3.4......3.2......3.2......1.0......3.2......1.2
..2.1.0....6.5.6....2.3.2....2.1.0....2.3.2....2.1.0....2.3.2....2.3.2
..1.0.1.0..5.6.7.6..3.4.3.2..1.2.1.0..1.2.1.0..1.2.1.0..3.4.3.2..3.4.3.4

Links

A194999 Number of lower triangles of a 3 X 3 0..n array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

2, 20, 42, 66, 90, 114, 138, 162, 186, 210, 234, 258, 282, 306, 330, 354, 378, 402, 426, 450, 474, 498, 522, 546, 570, 594, 618, 642, 666, 690, 714, 738, 762, 786, 810, 834, 858, 882, 906, 930, 954, 978, 1002, 1026, 1050, 1074, 1098, 1122, 1146, 1170, 1194, 1218
Offset: 1

Views

Author

R. H. Hardin, Sep 07 2011

Keywords

Comments

Row 3 of A194998.

Examples

			Some solutions for n=4:
..2......0......4......1......2......1......4......2......3......1......4
..3.2....1.2....3.2....2.1....1.2....2.3....3.4....1.2....4.3....2.3....3.2
..4.3.4..2.1.0..4.3.4..3.2.3..0.1.2..3.2.3..2.3.4..0.1.0..3.2.1..3.2.1..2.1.0

Links

Crossrefs

Cf. A194998.

Formula

Empirical: a(n) = 24*n - 30 for n>2.
Empirical g.f.: 2*x*(1 + 8*x + 2*x^2 + x^3) / (1 - x)^2. - Colin Barker, May 06 2018

A195000 Number of lower triangles of a 4 X 4 0..n array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

2, 80, 248, 458, 672, 888, 1104, 1320, 1536, 1752, 1968, 2184, 2400, 2616, 2832, 3048, 3264, 3480, 3696, 3912, 4128, 4344, 4560, 4776, 4992, 5208, 5424, 5640, 5856, 6072, 6288, 6504, 6720, 6936, 7152, 7368, 7584, 7800, 8016, 8232, 8448, 8664, 8880, 9096
Offset: 1

Views

Author

R. H. Hardin, Sep 07 2011

Keywords

Comments

Row 4 of A194998.

Examples

			Some solutions for n=4:
..1........4........3........4........2........1........1........0
..2.1......3.2......2.3......3.2......3.4......0.1......2.1......1.2
..1.0.1....2.1.2....3.4.3....2.1.2....2.3.4....1.2.1....3.2.3....2.3.2
..0.1.0.1..1.2.3.4..2.3.2.1..1.0.1.0..3.4.3.2..2.1.2.1..2.3.2.3..3.2.1.0

Links

Crossrefs

Cf. A194998.

Formula

Conjectures from Colin Barker, May 06 2018: (Start)
G.f.: 2*x*(1 + 38*x + 45*x^2 + 21*x^3 + 2*x^4 + x^5) / (1 - x)^2.
a(n) = 24*(9*n - 17) for n>4.
a(n) = 2*a(n-1) - a(n-2) for n>6.
(End)

A195001 Number of lower triangles of a 5 X 5 0..n array with each element differing from all of its horizontal and vertical neighbors by one.

Original entry on oeis.org

2, 576, 2290, 4990, 7858, 10804, 13754, 16706, 19658, 22610, 25562, 28514, 31466, 34418, 37370, 40322, 43274, 46226, 49178, 52130, 55082, 58034, 60986, 63938, 66890, 69842, 72794, 75746, 78698, 81650, 84602, 87554, 90506, 93458, 96410, 99362
Offset: 1

Views

Author

R. H. Hardin, Sep 07 2011

Keywords

Comments

Row 5 of A194998.

Examples

			Some solutions for n=4:
..0..........0..........1..........0..........0..........4..........1
..1.2........1.2........2.1........1.2........1.0........3.2........2.1
..2.1.0......2.1.2......1.2.3......2.1.2......2.1.0......2.3.2......1.2.3
..1.2.1.2....1.2.1.2....0.1.2.3....3.2.3.4....3.2.1.2....3.4.3.2....0.1.2.3
..0.1.2.3.4..2.1.0.1.0..1.2.3.2.3..2.3.2.3.2..2.3.2.1.0..4.3.2.1.0..1.2.3.4.3

Links

Crossrefs

Cf. A194998.

Formula

Empirical: a(n) = 2952*n - 6910 for n>6.
Empirical g.f.: 2*x*(1 + 286*x + 570*x^2 + 493*x^3 + 84*x^4 + 39*x^5 + 2*x^6 + x^7) / (1 - x)^2. - Colin Barker, May 06 2018
Showing 1-10 of 12 results. Next

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