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 1-4 of 4 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

A279160 Irregular triangle read by rows in which row n lists the perimeters of the parts of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 8, 6, 6, 16, 8, 8, 24, 10, 10, 32, 12, 8, 12, 20, 20, 14, 14, 48, 16, 16, 26, 26, 18, 16, 18, 64
Offset: 1

Views

Author

Omar E. Pol, Dec 07 2016

Keywords

Examples

			4;
8;
6, 6;
16;
8, 8;
24;
10, 10;
32;
12, 8, 12;
20, 20;
14, 14;
48;
16, 16;
26, 26;
18, 16, 18;
64;
...
Illustration of the 9th row:
.         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, so the 9th row of triangle is 12, 8, 12.

Crossrefs

Twice A278447.
Row sums give A274919.
Cf. A237271 gives the row lengths.
Cf. A000203, A196020, A235791, A236104, A237270, A237591, A237593, A245092, A262626.

A362817 Irregular triangle read by rows: T(n,k) (n>=1, k>=1) is the number of edges of the k-th polygon (or part), from left to right, of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 4, 4, 10, 4, 4, 12, 4, 4, 14, 4, 6, 4, 8, 8, 4, 4, 18, 4, 4, 8, 8, 4, 12, 4, 22, 4, 4, 22, 4, 4, 22, 4, 8, 8, 4, 8, 8, 4, 4, 26, 4, 10, 4, 8, 8, 4, 8, 8, 4, 28, 4, 4, 30, 4, 4, 30
Offset: 1

Views

Author

Omar E. Pol, May 04 2023

Keywords

Comments

Row n is [4, 4] if and only if n is an odd prime.
If the symmetric representation of sigma(n) has only one polygon (or part), or in other words, if n is a member of A174973 (also of the same sequence A238443) then row n has only a term: T(n,1) = 2 + 2*(A003056(n-1) + A003056(n)). Note that A174973 = A238443 also include all powers of 2 and all even perfect numbers.

Examples

			Triangle begins:
   4;
   6;
   4,  4;
  10;
   4,  4;
  12;
   4,  4;
  14;
   4,  6,  4;
   8,  8;
   4,  4;
  18;
   4,  4;
   8,  8;
   4, 12,  4;
  ...
Illustration of row 9:
         4
     _ _ _ _ _
    |_ _ _ _ _|
              |_ _ 6
              |_  |
                |_|_ _
                    | |
                    | |
                    | |  4
                    | |
                    |_|
.
For n = 9 the symmetric representation of sigma(9) has three parts from left to right as follows: a rectangle, a concave hexagon and a rectangle. The number of edges of the polygons are 4, 6, 4 respectively, so the row 9 of the triangle is [4, 6, 4].

Crossrefs

Row lengths give A237271.
Row sums give A362818.
Cf. A000079, A000203, A000396, A003056, A065091, A174973, A196020, A235791, A236104, A237270, A237591, A237593, A238443, A244363, A245092, A262626, A274919, A348705.

A362818 Total number of edges of all polygons (or parts) of the symmetric representation of sigma(n).

Original entry on oeis.org

4, 6, 8, 10, 8, 12, 8, 14, 14, 16, 8, 18, 8, 16, 20, 22, 8, 22, 8, 22, 24, 16, 8, 26, 18, 16, 24, 28, 8, 30, 8, 30
Offset: 1

Views

Author

Omar E. Pol, May 04 2023

Keywords

Comments

a(n) = 8 if and only if n is an odd prime.
If the symmetric representation of sigma(n) has only one polygon (or part), or in other words, if n is a member of A174973 (also of the same sequence A238443) then a(n) = 2 + 2*(A003056(n-1) + A003056(n)). Note that A174973 = A238443 also include all powers of 2 and all even perfect numbers.

Examples

			Illustration of a(9) = 14:
         4
     _ _ _ _ _
    |_ _ _ _ _|
              |_ _ 6
              |_  |
                |_|_ _
                    | |
                    | |
                    | |  4
                    | |
                    |_|
.
For n = 9 the symmetric representation of sigma(9) has three parts from right to left as follows: a rectangle, a concave hexagon and a rectangle. The number of edges of the polygons are 4, 6, 4 respectively, therefore the total number of edges is 4 + 6 + 4 = 14, so a(9) = 14.

Crossrefs

Row sums of A362817.
Cf. A000079, A000203, A000396, A003056, A065091, A174973, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A238443, A244363, A245092, A262626, A274919, A348705.
Showing 1-4 of 4 results.

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