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Sequence A002061, Hogben's Centered Polygonal Numbers    

This sequence is Sloane's A002061 and my MCS62032. It has a very simple formula, and a lot of nifty properties shown below. It has been in the OEIS since the books (See [4] and [4]), neither of which provided an explanation of why it is called "central polygonal numbers" For many years I was unable to work out any geometric interpretation of the sequence. Over the years, the OEIS (on-line database that evolved from the books) provided plenty of formulas and descriptions, but still nothing particularly "central polygonal".

In 2024 a reader informed me of an article that had been published pn arXiv: Paul Barry, "Centered polygon numbers, heptagons and nonagons, and the Robbins numbers", at arxiv.org/abs/2104.01644v2. The article shows that the sequences 1, 4, 10, 19, ... (Sloane's A005448) and 1, 5, 13, 25, ... (Sloane's A001844) correspond to the following arrangement of concentric triangles (on the left) and squares (on the right):

o / \ o---o---o---o / o \ | o---o---o | o / \ o o | o---o | o / / o \ \ | o | o | o | / o /o\ o \ o | o---o | o o / o---o \ o | o---o---o | / o----o----o \ o---o---o---o o----o-----o----o 1+4+8+12+... 1+3+6+9+... A001844 A005448

These are the "centered polygons" referenced in the name of sequence A002061. However, the concentric "polygons" in the case of A002061 have just two sides. Some other ways to interpret A002061 geometrically are given below.

The sequence begins: A0=1, A1=1, A2=3, A3=7, A4=13, A5=21, A6=31; and continues: 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, ...

The simplest direct formula for a term in the sequence is AN=N2-N+1; the iterative formula is AN+1=AN+2N. Many simple descriptions of the sequence, including those given below, result from these formulas or slight variations.

The sequence is given by Sloane in his first book [2] with a reference to Hogben [1],[3]; in the second book [4] he uses it as an example, the same references and again no explanation for the name. Since the formula is given, and would be easy enough to find by the techniques (such as difference analysis) given in the book, perhaps Sloane thought the name was not important, just an arbitrary designation that fully serves its purpose so long as it is unique.

If you look at other "figurate numbers" such as the triangular numbers (1, 3, 6, 10, 15, 21, ...: A000217), the squares, tetrahedral numbers (1, 4, 10, 20, 35, ...: A000292) and so on, there are simple geometric interpretations for the numbers. For example, for the hexagonal numbers (1, 7, 19, 37, 61, ...: A003215) you can think of circles or circular objects (try small coins) arranged in a hexagonal pattern: start with one coin, then add 6 around it, just touching to form a hexagonal pattern of 7 coins. Then add 12 more to make a bigger hexagon, then 18 more and so on. The other figurate numbers work in a similar way: always starting with one, and adding 3, 5, 7, 9 in turn (to get the squares) or 2, 3, 4, 5 (to get the triangular numbers), or the hexagonal numbers 7, 19, 37, 61 (to get the cubes), and so on.

What does this have to do with our sequence?

The sequence in question has a similar formula: starting with 1, add the even numbers 2, 4, 6, 8 in turn to create the sequence. But there is no obvious "polygonal" shape to justify the name "centered polygonal numbers". For good examples of polygonal shapes, see my pages for A006542 and A094534.

Now try making shapes for this sequence, with coins: put one coin down, then put 2 coins with it to make three; next add 4 to make 7; then add 6 to make 13, and so on. You will be hard pressed to find a way to make it fit any consistent "polygonal" shape. Here for example is a contrived attempt using a hexagonal grid:

+ + + + o o + + + + + + + o o o + + + + + o o + o o o o + o o o o + now o o + o + + o o o + o o o o o o o o o o + + o o + o o + o o o o o o o o what? + + + o o o    1 3 7 13 21 31 43

Rows of Odd Numbers

However there is a nice geometric pattern in two dimensions that comes out of the formulas shown below. Arrange the odd numbers in successive rows. Group the rows as shown below (starting with 1 row, then 2, then 3, and so on). Notice that all numbers AN in our sequence appear as the first row in a group of N rows:

1 o

3 o o o
5 o o o o o

7 o o o o o o o
9 o o o o o o o o o
11 o o o o o o o o o o o

13 o o o o o o o o o o o o o
15 o o o o o o o o o o o o o o o
17 o o o o o o o o o o o o o o o o o
19 o o o o o o o o o o o o o o o o o o o

21 o o o o o o o o o o o o o o o o o o o o o
23 o o o o o o o o o o o o o o o o o o o o o o o
25 o o o o o o o o o o o o o o o o o o o o o o o o o
27 o o o o o o o o o o o o o o o o o o o o o o o o o o o
29 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o
...

Reader Rui Pinto showed me the most elegant way to get the sequence, which results from writing every positive integer in an isosceles triangle shape:

row sums 1 1 = 0^3+1^3 2 3 4 9 = 1^3+2^3 5 6 7 8 9 35 = 2^3+3^3 10 11 12 13 14 15 16 91 = 3^3+4^3 17 18 19 20 21 22 23 24 25 189 = 4^3+5^3 26 27 28 29 30 31 32 33 34 35 36 341 = 5^3+6^3 37 38 39 40 41 42 43 44 45 46 47 48 49 559 = 6^3+7^3 50 51 . . . (etc...) Rui Pinto's wide isosceles triangle arrangement

and the sequence runs down the center column. Also, the rows each have an odd number of items, like the rows of o's in the previous figure — and the rightmost number in each row is a square N2. The odds and evens are segregated into alternating columns, and the sum of all the numbers in each row is the sum of two consecutive cubes.

Alternately, you can make two different narrower types of triangle with just the odd numbers. The one on the right is the above triangle with the even columns removed:

sum 1 1 1 3 5 8 3 7 9 11 27 5 7 9 13 15 17 19 64 11 13 15 21 23 25 27 29 125 17 19 21 23 25 31 33 35 37 39 41 216 27 29 31 33 35 43 45 47 49 51 52 55 343 37 39 41 43 45 47 49 57 59 61 . . . (etc...) 51 53 55 57 . . . (etc.) equilateral triangle arrangement "narrow isosceles" arrangement

Once again, the squares N2 (for odd N only, but with gaps where the even ones would fall) are lined up in an easily identifiable pattern.

The sum of each group of lines of o symbols in the first figure above, or the row sums of the equilateral triangle, gives the cubes N3. This is shown by the following figure that rearranges and regroups the rows into N squares:

1 o    8 o o o * o o o o .    27 o o o o o o o * * o o o o o o o o o o o o o o o o o o . .    64 o o o o o o o o o o o o o * * * o o o o o o o o o o o o o o o * o o o o o o o o o o o o o o o o . o o o o o o o o o o o o o o o o . . .    125 o o o o o o o o o o o o o o o o o o o o o * * * * o o o o o o o o o o o o o o o o o o o o o o o * * o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o . . o o o o o o o o o o o o o o o o o o o o o o o o o . . . .

In each case, a number of symbols marked "." have been moved to new positions "*" to make the last square complete. The number of symbols that needs to be moved is: 0, 1, 2, 4, 6, 9, 12, 16, 20, ...; itself a figurate sequence consisting of the squares X2 alternating with the almost-square rectangles X(X+1), and defined by the iteration BN=B(N-2)+N.

Relation to Kissing Spheres

There is another even more elegant geometrical interpretation, but it diverges even further from the name's suggestion "polygonal". Consider a unit sphere in N dimensions. How many other unit spheres may touch it, without intersecting it or each other? The total, counting the central sphere, must be at least AN+1 (proving that it is exactly equal to AN+1, for dimensions of 3 and higher, is notoriously difficult). 0 dimensions is a minimal case, the entire universe is one point, A1=1. In 1 dimension a sphere is a line segment, with two line segments touching it on either side: A2=3. In 2 dimensions we have the hexagonal arrangement of 7 coins mentioned above, which can be thought of as a one-dimensional row of 3, with 2 more above and 2 more below: 2+3+2=A3=7. In 3 dimensions, we have 7 spheres arranged in this manner, with a triangle of 3 above and 3 more below: 3+7+3=A4=13. For 4 dimensions, along one hyperplane are 13 hyperspheres arranged in the pattern just described, then along two parallel hyperplanes on either side are two tetrahedral arrangements of 4 hypershperes each: 4+13+4=A5=21. A similar arrangement in 5 dimensions, with a hyperplane of 21 "spheres" flanked by two hyperplanes with 4 "spheres" each gives 5+21+5=31, and so on.

Given this interpretation, the sequence gives a sequence of figurate numbers of progressively higher dimensionality. Each is a sort of "centered polyhedron" on N dimensions, with N increasing by 1 each time. It is also important to note that (starting with 4 dimensions), the number given by this sequence is not the maximum possible number of hyperspheres that can touch a central hypersphere. In 4 dimensions you can actually get 25 (the central hypersphere plus 24), and progressively larger numbers in higher dimensions. (This sequence, called the "kissing number sequence", is very difficult to determine, but it is known that for dimensions 1 to 9, it is at least 2, 6, 12, 24, 40, 72, 126, 240, 272: Sloane's A002336)

Amarnath Murthy found the relation (N-1)AN + 1 = N3, but erroneously reported that one must "drop the "first three terms" for the relation to hold. It actually works for all of the terms, and expects the sequence to be numbered so that the N and AN go like this:

N AN (N-1)AN + 1 N3
-1 1 -2×1 + 1 -1
0 1 -1×1 + 1 0
1 3 0×3 + 1 1
2 7 1×7 + 1 8
3 13 2×13 + 1 27
4 21 3×21 + 1 64
etc.

In the OEIS the sequence is defined A0=1, A1=1, A2=3, A3=7, and so on, and Murthy's formula needs to be changed accordingly:

N AN (N-2)AN + 1 (N-1)3
0 1 -2×1 + 1 -1
1 1 -1×1 + 1 0
2 3 0×3 + 1 1
3 7 1×7 + 1 8
4 13 2×13 + 1 27
5 21 3×21 + 1 64
etc.

Perhaps the best alteration of Murthy's formula is:

(N-1)AN+1 + 1 = N3

because it is so similar to:

(N-1)(N+1) + 1 = N2


Some other sequences are discussed here.


References

[1] L. Hogben, Choice and Chance by Cardpack and Chessboard, v. 1, Chanticleer Press, New York, 1950.

[2] Neil James Alexander Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (sequence 1049).

[3] R. Honsberger, Ingenuity in Mathematics, Random House, New York, 1970.

[4] Neil James Alexander Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (page 31 and sequence M2638).

[5] Paul Barry, "Centered polygon numbers, heptagons and nonagons, and the Robbins numbers", on arXiv: arxiv.org/abs/2104.01644v2.



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