A244362 -id:A244362 - cp's OEIS frontend

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).

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

Omar E. Pol, Jun 26 2014

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

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

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

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

A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

Original entry on oeis.org

8, 24, 48, 80, 112, 160, 200, 264, 328, 408, 464, 560, 624, 728, 832, 960, 1040, 1184, 1272, 1432, 1576, 1728, 1832, 2024, 2160, 2336, 2512, 2736

Offset: 1

Omar E. Pol, Jun 26 2014

Partial sums of A244371.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).

			Illustration of the structure after 16 stages (Contains 960 toothpicks):
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.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244371, A244970, A244971, A245092.

a(n) = 4*A244362(n) = 8*A244360(n).

a(8) corrected and more terms from Omar E. Pol, Oct 18 2014

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

Omar E. Pol, Jun 26 2014

Partial sums of A244361.

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

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

More terms from Omar E. Pol, Oct 18 2014

A244361 Number of toothpicks added at 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, 2, 3, 4, 4, 6, 5, 8, 8, 10, 7, 12, 8, 13, 13, 16, 10, 18, 11, 20, 18, 19, 13, 24, 17, 22, 22, 28, 16, 30, 17, 32

Offset: 1

Omar E. Pol, Jun 26 2014

Partial sums give A244360. - Omar E. Pol, Oct 18 2014

Cf. A000203, A237270, A237271, A237591, A237593, A244360, A244362, A244363, A244370, A244371, A279228.

a(n) = A244363(n)/2.

a(n) = A244371(n)/8. - Omar E. Pol, Oct 18 2014

a(13)-a(28) from Omar E. Pol, Oct 18 2014

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

cp's OEIS Frontend

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).

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A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

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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).

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A244361 Number of toothpicks added at 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).

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