cp's OEIS Frontend

This sequence is related to A045944 by a(n) = n*A045944(n)-Sum_{i=0..n-1} A045944(i); this is the case d=6 in the identity n^2*(d*n+d-2)/2 - sum(k*(d*k+d-2)/2, k=0..n-1) = n*(n+1)*(2*d*n+d-3)/6 . - Bruno Berselli, Nov 19 2010

Bisection (even part) of A002717. See the Conway and Guy reference. - Wolfdieter Lang, Apr 16 2020

Table starts

...........2.............3............4.........5..........6......7...8.9

...........2............10...........28........60........110....182.280

..........38...........624.........4740.....23036......83994.250668

.........838........173860......7788648.147823100.1631287078

.......75302.....421158558.189412482924

....27842760.8496020082856

.37121438930

There are 2^(n^2) binary n X n matrices (entries of {0,1}). There are 2^(n*(n+1)/2) symmetric binary matrices. There are A184948(n,k) symmetric binary n X n matrices with k ones.

This sequence is the triangle T(n,k) of symmetric binary n x n matrices with k ones but no 2 X 2 submatrix with all entries = 1. [So in the display of these matrices there is no rectangle with four 1's at the corners.]

The row lengths minus 1 are 0, 1, 3, 6, 9, 12, 17, 21, 24, 29, ... and indicate the maximum number of 1's than can be packed into a symmetric binary n X n matrix without creating an all-1 quadrangle/submatrix of order 2.

Define the "entropy" (or variance) of a permutation pi to be Sum_{i=1..n} (pi(i)-i)^2 = A006331(n) - 2*Sum_i i*pi(i), as in A126972.

This characteristic is obviously an even number, 2*v(pi).

Row n of the triangle shows the statistics (frequency distribution) of v for the n! = A000142(n) possible permutations of [n].

T(n,0)=1 arises the identity permutation where v=0.

T(n,1)=n-1 arises from the n-1 different ways of creating an entropy of 2 by swapping a pair of adjacent entries in the identity permutation.

The final 1 in each row arises from the permutation with maximal entropy, that is the permutation with integers reversed relative to the identity permutation.

Row n has 1+A000292(n-1) entries. Row sums are sum_{v=0..A000292(n-1)} T(n,v) = n!.

Removing zeros in A135298 creates a sequence which is similar in the initial terms, because contributions to A135298(n) stem from permutations of some unique [j] if n is not too large, which establishes a 1-to-1 correspondence between the term A006331(n)-2*sum_i i*pi(i) mentioned above and the defining formula in A135298.

The rows of this triangle have a geometric interpretation. Let P_n be the n-dimensional permutohedron, the Voronoi cell of the lattice A_n* (Conway-Sloane, 1993, p. 474), which is a polytope with (n+1)! vertices. Start at any vertex, and count how many vertices there are at squared-distance v from the starting vertex: this is T(n+1,v). For example, in three dimensions the permutohedron is a truncated octahedron, the squared distances from a vertex to all the vertices are (when suitably scaled) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and the numbers of vertices at these distances are 1, 3, 1, 4, 2, 2, 2, 4, 1, 3, 1, which is row 4 of the array. See Chap. 21, Section 3.F, op. cit., for further details. - N. J. A. Sloane, Oct 13 2015

A member of the accumulation chain ... < A185879 < A185877 < A185878 < A185880 < ...

See A144112 for the definition of accumulation array.

Hankel transform of A137398(n+1) (conjecture).

The sequence starts with a central vertex and expands outward with (n-1) centered polygonal pyramids producing a truncated tetrahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon in each face. For centered triangles see A005448 and centered hexagons A003215.

This sequence is the 18th in the series (1/12)*t*(2*n^3-3*n^2+n)+2*n-1 for t = 2, 4, 6, ... gives A049480, A005894, A063488, A001845, A063489, A005898, A063490, A057813, A063491, A005902, A063492, A005917, A063493, A063494, A063495, A063496 and t = 36. While adjusting for offsets, the beginning sequence A049480 is generated by adding the square pyramidal numbers A000330 to the odd numbers A005408 and each subsequent sequence is found by adding another set of square pyramidals A000330. (T/2) * A000330(n) + A005408(n). At 30 * A000330 + A005408 = centered dodecahedral numbers, 36 * A000330 + A005408 = A193228 truncated octahedron and 90 * A000330 + A005408 = A193248 = truncated icosahedron and dodecahedron. All five of the "Centered Platonic Solids" numbers sequences are in this series of sequences. Also 4 out of five of the "truncated" platonic solid number sequences are in this series. - Bruce J. Nicholson, Jul 06 2018

It would be good to have a detailed description of how the sequence is constructed. Maybe in the Examples section? - N. J. A. Sloane, Sep 07 2018

From Petros Hadjicostas, May 25 2019: (Start)

According to the Schoenflies notation used in Chemistry, the point group C_{2v} is the cyclic group C_2 "with the addition of" 2 "mirror planes containing the axis of rotation (vertical planes)" (see the Wikipedia article about the Schoenflies notation).

According to the Wikipedia article about Molecular symmetry, a simple description of typical geometry of this group is "angular (H2O) or see-saw (SF4)" (where SF4 = sulfur tetrafluoride).

According to Cyvin et al. (1996, p. 496), a catafusene is a "simply connected catacondensed polyhex". "It is a system consisting of congruent regular hexagons, where any two hexagons either share exactly one edge or are disjointed". "A tri-4-catafusene is generated by contraction of exactly three of its hexagons to tetragons" (see p. 499).

In general (see p. 496), "any alpha-q-catafusene (q = 3-5) can be generated from a catafusene on converting alpha of its hexagons to q-gons by contraction". Here, obviously, alpha = 3 and q = 4.

Here, a(n) is the number of (non-equivalent) unbranched tri-4-catafusenes under the symmetry point group C_{2v}, where n = number of polygons (also known as "rings") in the unbranched tri-4-catafusenes. (Hence, n - 3 is the total number of the remaining hexagons since 3 of the original hexagons became tetragons.) In the paper, the letter r is used to denote the number of polygons rather than the letter n (see p. 496).

The numbers (a(n): n >= 3) = (a(r): r >= 3) appear in Table 1 on p. 500 of Cyvin et al. (1996) under the point group C_{2v}.

In the paper, a(n) is denoted by M_r (p. 500) and is called the total number of "mirror-symmetrical (C_{2v}) systems" (here r = n). It is decomposed into the sum C_r + L_r, where C_r is the number of "centrosymmetrical (C_{2h}) systems" (p. 499) and L_r = (1/2)*binomial(r, 3) - (1/8)*(1-(-1)^r)*(r-1) (see Eq. (15) on p. 499). The numbers L_r count some kind of symmetry described on p. 499 of the paper, but it is unclear what they exactly count.

The point group C_{2h} mentioned above (with the centrosymmetrical systems) is the cyclic group C_2 "with the addition of a mirror (reflection) plane perpendicular to the axis of rotation (horizontal plane)" (see the Wikipedia articles about Molecular symmetry and the Schoenflies notation).

Unfortunately, the letter C is used for two different purposes in the paper: to denote various point groups in three dimensions and to denote the number of non-equivalent systems of contracted catafusenes under some kind of symmetry.

The numbers (C_r: r >= 0), which apparently appear in Table 1 (p. 500) under the point group C_{2h} (see above), satisfy C_{2*i} = 0 and C_{2*i+3} = (1/2)*Sum_{1 <= j <= i} j*a_{i+1,j} (see Eq. (17), p. 500), where the double array a_{i,j} appears on p. 498 (Section 5) of the paper. We have a_{i,j} = binomial(i-1, j-1)*2^(i-j) = A038207(i-1, j-1) for i, j >= 1 (see Eq. (32), p. 502 in the paper).

Putting the above information together, we can easily prove the formulas for a(n) that appear below in the FORMULA section.

Note that the notation for the current sequence a(n) should not be confused with the double array a_{i,j} defined on p. 498 of the paper.

(End)

k 1 2 3 4 5 6 7 8 9 10 11 12

---------------------------------------------------------------------------------

n 1| 1 2 6 24 120 720 5040 40320 362880 3628800 39916800 479001600

2| 3 10 54 402 3810 43776

3| 6 28 214 2348 32808

4| 10 60 594 8556

5| 15 110 1334

6| 21 182 2614

7| 28 280

8| 36 408

9| 45 570

10| 55 770