A237590 a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 18, 19, 21, 23, 26, 27, 29, 30, 32, 33, 37, 39, 41, 42, 45, 47, 51, 52, 54, 55, 57, 58, 62, 64, 67, 68, 70, 72, 76, 77, 79, 80, 82, 84, 87, 89, 91, 92, 95, 98, 102, 104, 106, 107, 111, 112, 116, 118, 120, 121, 123, 125, 130, 131, 135, 136, 138, 140, 144, 147, 149, 150, 152, 154

Offset

1,2

Comments

The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.

Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.

For more information see A237593 and A237270.

a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - Omar E. Pol, Apr 20 2016

Links

Robert Price, Table of n, a(n) for n = 1..5000

Omar E. Pol, An infinite stepped pyramid

Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)

Omar E. Pol, Perspective view of the stepped pyramid (first 16 levels)

Formula

a(n) = A317109(n) - A294723(n) + 1 (Euler's formula). - Omar E. Pol, Jul 21 2018

Example

Illustration of initial terms:
.                                                         _ _ _ _
.                                           _ _ _        |_ _ _  |_
.                               _ _ _      |_ _ _|       |_ _ _|   |_
.                     _ _      |_ _  |_    |_ _  |_ _    |_ _  |_ _  |
.             _ _    |_ _|_    |_ _|_  |   |_ _|_  | |   |_ _|_  | | |
.       _    |_  |   |_  | |   |_  | | |   |_  | | | |   |_  | | | | |
.      |_|   |_|_|   |_|_|_|   |_|_|_|_|   |_|_|_|_|_|   |_|_|_|_|_|_|
.
.
.       1      2        4          5            7              8
.
For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
The sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 33. On the other hand after 6 stages the sum of all parts of the diagram is [1] + [3] + [2+2] + [7] + [3+3] + [12] = 33, equaling the sum of all divisors of all positive integers <= 6.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
From Omar E. Pol, Dec 25 2020: (Start)
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
|_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
|_ _ _ _ _ _ _ _ _ _ _ _ _ _  |
|_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
|_ _ _ _ _ _ _ _ _ _ _ _ _  | |
|_ _ _ _ _ _ _ _ _ _ _ _ _| | |
|_ _ _ _ _ _ _ _ _ _ _ _  | | |_ _ _
|_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _  |
|_ _ _ _ _ _ _ _ _ _ _  | | |_ _  | |_
|_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_  |_
|_ _ _ _ _ _ _ _ _ _  | |       |_ _|   |_
|_ _ _ _ _ _ _ _ _ _| | |_ _    |_  |_ _  |_ _
|_ _ _ _ _ _ _ _ _  | |_ _ _|     |_  | |_ _  |
|_ _ _ _ _ _ _ _ _| | |_ _  |_      |_|_ _  | |
|_ _ _ _ _ _ _ _  | |_ _  |_ _|_        | | | |_ _ _ _ _ _
|_ _ _ _ _ _ _ _| |     |     | |_ _    | |_|_ _ _ _ _  | |
|_ _ _ _ _ _ _  | |_ _  |_    |_  | |   |_ _ _ _ _  | | | |
|_ _ _ _ _ _ _| |_ _  |_  |_ _  | | |_ _ _ _ _  | | | | | |
|_ _ _ _ _ _  | |_  |_  |_    | |_|_ _ _ _  | | | | | | | |
|_ _ _ _ _ _| |_ _|   |_  |   |_ _ _ _  | | | | | | | | | |
|_ _ _ _ _  |     |_ _  | |_ _ _ _  | | | | | | | | | | | |
|_ _ _ _ _| |_      | |_|_ _ _  | | | | | | | | | | | | | |
|_ _ _ _  |_ _|_    |_ _ _  | | | | | | | | | | | | | | | |
|_ _ _ _| |_  | |_ _ _  | | | | | | | | | | | | | | | | | |
|_ _ _  |_  |_|_ _  | | | | | | | | | | | | | | | | | | | |
|_ _ _|   |_ _  | | | | | | | | | | | | | | | | | | | | | |
|_ _  |_ _  | | | | | | | | | | | | | | | | | | | | | | | |
|_ _|_  | | | | | | | | | | | | | | | | | | | | | | | | | |
|_  | | | | | | | | | | | | | | | | | | | | | | | | | | | |
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
.
(End)

Mathematica

(* total number of parts in the first n symmetric representations *)
(* Function a237270[] is defined in A237270 *)
(* variable "previous" represents the sum from 1 through m-1 *)
a237590[previous_, {m_, n_}]:=Rest[FoldList[Plus[#1, Length[a237270[#2]]]&, previous, Range[m, n]]]
a237590[n_]:=a237590[0, {1, n}]
a237590[78] (* data *)
(* Hartmut F. W. Hoft, Jul 07 2014 *)

Crossrefs

Partial sums of A237271.

Compare with A060831 (analog for the diagram that contains subparts).

Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.

Sequence in context: A022768 A229973 A007951 * A207336 A213539 A190849

Adjacent sequences: A237587 A237588 A237589 * A237591 A237592 A237593

Keyword

nonn

Author

Omar E. Pol, Mar 31 2014

Extensions

Definition clarified by Omar E. Pol, Jul 21 2018

Status

approved