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 1-7 of 7 results.

A244363 Number of toothpicks added at n-th stage in the toothpick structure of the symmetric representation of sigma of the first n positive integers in the first quadrant (without the axis x and y).

Original entry on oeis.org

2, 4, 6, 8, 8, 12, 10, 16, 16, 20, 14, 24, 16, 26, 26, 32, 20, 36, 22, 40, 36, 38, 26, 48, 34, 44, 44, 56, 32, 60, 34, 64
Offset: 1

Views

Author

Omar E. Pol, Jun 26 2014

Keywords

Comments

Partial sums give A244362. - Omar E. Pol, Oct 18 2014
a(n) is also the sum of semiperimeters of the parts of the symmetric representation of sigma(n). - Omar E. Pol, Dec 11 2016
It appears that a(n) is also the total length of the horizontal cuts that must be made at level n, starting from the top, in the diagram of the "isosceles triangle shaped" 4*n-gon described in A237593 to transform it into a pop-up card which when folded 90 degrees has the property that the total area of its holes at level n is equal to A000203(n). Note that the pop-up card has essentially the same structure as the stepped pyramid described in A245092. The holes of the pop-up card are equivalent to the terraces of the stepped pyramid, therefore both objects share many properties. - Omar E. Pol, Mar 08 2023

Crossrefs

Cf. A000203, A196020, A237270, A237271, A237593, A244360, A244361, A244362, A244370, A244371, A245092, A262626, A274919.

Formula

a(n) = 2*A244361(n).
a(n) = A244371(n)/4. - Omar E. Pol, Oct 18 2014
a(n) = A274919(n)/2. - Omar E. Pol, Dec 11 2016

Extensions

a(13)-a(28) from Omar E. Pol, Oct 18 2014
Definition clarified by Omar E. Pol, Mar 08 2023
a(29)-a(32) from Omar E. Pol, May 04 2023

A244371 Number of toothpicks added at n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

Original entry on oeis.org

8, 16, 24, 32, 32, 48, 40, 64, 64, 80, 56, 96, 64, 104, 104, 128, 80, 144, 88, 160, 144, 152, 104, 192, 136, 176, 176, 224, 128, 240, 136, 256
Offset: 1

Views

Author

Omar E. Pol, Jun 26 2014

Keywords

Comments

Partial sums give A244370.

Crossrefs

Cf. A000203, A196020, A236104, A237270, A237271, A237591, A237593, A244360-A244363, A244370, A244371, A244970, A244971, A245092.

Formula

a(n) = 4*A244363(n) = 8*A244361(n). - Omar E. Pol, Oct 16 2014

Extensions

a(13)-a(28) from Omar E. Pol, Oct 18 2014
a(29)-a(32) from Omar E. Pol, May 04 2023

A244360 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of half sigma in the first octant (without the axis x and without the main diagonal).

Original entry on oeis.org

1, 3, 6, 10, 14, 20, 25, 33, 41, 51, 58, 70, 78, 91, 104, 120, 130, 148, 159, 179, 197, 216, 229, 253, 270, 292, 314, 342
Offset: 1

Views

Author

Omar E. Pol, Jun 26 2014

Keywords

Comments

Partial sums of A244361.

Crossrefs

Cf. A000203, A237270, A237271, A237593, A244361, A244362, A244363, A244370, A244371.

Formula

a(n) = A244362(n)/2 = A244370(n)/8. - Omar E. Pol, Oct 18 2014

Extensions

More terms from Omar E. Pol, Oct 18 2014

A279228 Number of unit steps that are shared by the smallest and largest Dyck path of the symmetric representation of sigma(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 4, 0, 2, 0, 8, 0, 10, 2, 4, 0, 14, 0, 16, 0, 6, 6, 20, 0, 16, 8, 10, 0, 26, 0, 28, 0
Offset: 1

Views

Author

Omar E. Pol, Dec 08 2016

Keywords

Comments

a(n) is also the number of unit steps that are shared by the largest Dyck path of the symmetric representation of sigma(n) and the largest Dyck path of the symmetric representation of sigma(n-1), in a quadrant of the square grid.
For more information about the Dyck paths of the symmetric representation of sigma(n) see A237593.

Examples

			Illustration of initial terms (n = 1..12) using the spiral described in A239660:
.               _ _ _ _ _ _
.              |  _ _ _ _ _|_ _ _ _ _
.         0   _| |         |_ _ _ _ _|
.           _|_ _|                   |_ _ 2
.       _ _| |      _ _ _ _          |_  |
.      |  _ _|  0 _|  _ _ _|_ _ _      |_|_ _
.      | |      _|   |     |_ _ _|  2      | |
.      | |     |  _ _|           |_ _      | |
.      | |     | |    0 _ _        | |     | |
.      | |     | |     |  _|_ 0    | |     | |
.     _|_|    _|_|    _|_| |_|    _|_|    _|_|    _
.    | |     | |     | |         | |     | |     | |
.    | |     | |     |_|_ _     _| |     | |     | |
.    | |     | |      0|_ _|_ _|  _|     | |     | |
.    | |     |_|_          |_ _ _|0   _ _| |     | |
.    | |         |_                 _|  _ _|     | |
.    |_|_ _     4  |_ _ _ _        |  _|    _ _ _| |
.          |_      |_ _ _ _|_ _ _ _| |  0 _|    _ _|
.            |_            |_ _ _ _ _|  _|     |
.         8    |                       |      _|
.              |_ _ _ _ _ _            |  _ _|
.              |_ _ _ _ _ _|_ _ _ _ _ _| |      0
.                          |_ _ _ _ _ _ _|
.
.
For an illustration of the following examples see the last lap of the above spiral starting in the first quadrant.
For n = 9 the Dyck paths of the symmetric representation of sigma(9) share 2 unit steps, so a(9) = 2.
For n = 10 the Dyck paths of the symmetric representation of sigma(10) meet at the center, but they do not share unit steps, so a(10) = 0.
For n = 11 the Dyck paths of the symmetric representation of sigma(11) share 8 unit steps, so a(11) = 8.
For n = 12 the Dyck paths of the symmetric representation of sigma(12) do not share unit steps, so a(12) = 0.
Note that we can find the spiral on the terraces of the stepped pyramid described in A244050.

Crossrefs

Cf. A279029 gives the indices of the zero values.
Cf. A279244 gives the indices of the positive values.
Cf. A000203, A005843, A008586, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239660, A239931-A239934, A244050, A244361, A244363, A245092, A262626, A348705.

Formula

a(n) = 2*n - A244363(n) = 2*(n - A244361(n)).
a(n) = A008586(n) - A348705(n). - Omar E. Pol, Dec 13 2021

A244362 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the first quadrant (without the axes x and y).

Original entry on oeis.org

2, 6, 12, 20, 28, 40, 50, 66, 82, 102, 116, 140, 156, 182, 208, 240, 260, 296, 318, 358, 394, 432, 458, 506, 540, 584, 628, 684
Offset: 1

Views

Author

Omar E. Pol, Jun 26 2014

Keywords

Comments

Partial sums of A244363.

Examples

			Illustration of the structure after 7 stages (Contains 50 toothpicks):
.
.    _ _ _ _
.    _ _ _ _|
.    _ _ _  |_
.    _ _ _|   |_ _
.    _ _  |_ _  | |
.    _ _|_  | | | |
.    _  | | | | | |
.     | | | | | | |
.
.     1 2 3 4 5 6 7
.

Crossrefs

Cf. A000203, A237270, A237271, A237593, A244360, A244361, A244363, A244370, A244371.

Formula

a(n) = 2*A244360(n).
a(n) = A244370(n)/4. - Omar E. Pol, Oct 18 2014

Extensions

More terms from Omar E. Pol, Oct 18 2014

A274919 Sum of all perimeters of all parts of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 8, 12, 16, 16, 24, 20, 32, 32, 40, 28, 48, 32, 52, 52, 64, 40, 72, 44, 80, 72, 76, 52, 96, 68, 88, 88, 112, 64, 120, 68, 128
Offset: 1

Views

Author

Omar E. Pol, Dec 11 2016

Keywords

Comments

a(n) is also the number of toothpicks added at n-th stage in the toothpick structure of the symmetric representation of sigma in two quadrants (without the axis x and y).

Examples

			Illustration of a(9) = 32:
.         12
.     _ _ _ _ _
.    |_ _ _ _ _|
.               _ _ 8
.              |_  |
.                |_|  _
.                    | |
.                    | |
.                    | |  12
.                    | |
.                    |_|
.
For n = 9 the symmetric representation of sigma(9) = 13 has three parts of areas 5, 3, 5 respectively. The perímeters of the parts are 12, 8 and 12 as shown above. The sum of the perimeters is 12 + 8 + 12 = 32, so a(9) = 32.

Crossrefs

Cf. A000203, A008586, A196020, A236104, A237270, A237271, A237593, A244361, A244363, A244370, A244371, A245092, A262626, A279228.

Formula

a(n) = 4*A244361(n) = 2*A244363(n) = A244371(n)/2.
a(n) = A008586(n) - 2*A279228(n). - Omar E. Pol, May 04 2023

Extensions

a(29)-a(32) from Omar E. Pol, May 04 2023

A244250 Triangle read by rows in which row n lists the widths in the first octant of the symmetric representation of sigma(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Omar E. Pol, Oct 26 2014

Keywords

Comments

For the definition of k-th width of the symmetric representation of sigma(n) see A249351.
Row n list the first n terms of the n-th row of A249351.
It appears that the leading diagonal is also A067742 (which was conjectured by Michel Marcus in the entry A237593 and checked with two Mathematica functions up to n = 100000 by Hartmut F. W. Hoft).
For more information see A237591, A237593.

Examples

			Triangle begins:
1;
1, 1;
1, 1, 0;
1, 1, 1, 1;
1, 1, 1, 0, 0;
1, 1, 1, 1, 1, 2;
1, 1, 1, 1, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1;
1, 1, 1, 1, 1, 0, 0, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0;
1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2;
1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0;
...

Crossrefs

Cf. A000203, A067742, A071562, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A239660, A239932-A239934, A240542, A241008, A241010, A244360, A244361, A245685, A246955, A246956, A247687, A249223, A249351, A250068, A250070, A250071.
Showing 1-7 of 7 results.

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