Rajan Murthy - cp's OEIS frontend

A352622 Number of regular convex polytopes that can be formed with n indistinguishable points located at the vertices, coinciding in equal frequency at each vertex, if coinciding at all.

1, 2, 2, 4, 3, 6, 3, 8, 4, 7, 3, 12, 3, 7, 6, 12, 3, 11, 3, 13, 6, 7, 3, 20, 5, 7, 6, 12, 3, 16, 3, 16, 6, 7, 7, 20, 3, 7, 6, 20, 3, 16, 3, 12, 10, 7, 3, 27, 5, 12, 6, 12, 3, 16, 7, 19, 6, 7, 3, 29, 3, 7, 10, 20, 7, 16, 3, 12, 6, 17, 3, 31, 3, 7, 10, 12, 7, 16

Offset: 1

Rajan Murthy, Mar 24 2022

nonn

For n = 1: there is only a 0-dimensional simplex.
For n = 2: the two points may coincide or may form a 1-dimensional simplex.
For n = 3: the three points may coincide or may form a 2-dimensional simplex.
For n = 2^(k+1), where k is a positive integer: a(n) = k + (k+2) + (k-1) + (k-1) = 4*k: k polygons (one for each factor > 2), k+2 simplexes (one for each factor), k-1 cubes (one for each even factor > 4, the cubes for 2 and 4 are a simplex and polygon, respectively), and k-1 orthoplexes (one for each even factor > 4, orthoplexes with 1, 2, and 4 vertices are already counted).
For prime numbers greater than 3 (n = p > 3, where p is prime): a(n) is always 3:
(1) the 0-dimensional polytope (all points coinciding), (2) a 2-dimensional p-gon, where p is a prime n, and (3) a (p-1)-dimensional simplex.
For even numbers which are not powers of 2: a(n) = 2*(number of factors) + (number of even factors) - 3 + adjustments. The adjustments are as follows: -1 if n is a multiple of 3; -1 if n is a multiple of 4; +1 for each positive integer k such that 2^(k+2) is a factor of n; +1 for each factor of n which is in the set (12,20,24,120,600). With the exception of factors 1 and 2, every factor contributes a simplex and a polygon. Even factors add a third polytope which is an orthoplex. Factors 1 and 2 only add a zero-dimensional and one-dimensional simplex respectively and so a total of three is subtracted (-1 for each of factors 1 + 2 and -1 for the even factor 2). The polygon and the simplex to which the factor of 3 maps are identical leading to an adjustment of -1. The polygon and the 2-dimensional "cube" that a factor of 4 maps to are identical also leading to a -1 adjustment. Factors which are powers of 2 greater than 4 and factors which correspond to a polytope peculiar to 3 or 4 dimensions each add one more possible polytope.
For nonprime odd numbers which are multiples of 3: a(n) = 2*(the number of factors) - 2. Each factor maps to a polygon and a simplex, but for the factor 3 the polygon is the simplex, and the factor 1 maps to a single coincident point.
For nonprime odd numbers which are not multiples of 3: a(n) = 2*(the number of factors) - 1. Each factor > 1 maps to a polygon and a simplex and the factor 1 maps to a single coincident point.

			For n = 12, the set of factors of 12 is (1, 2, 3, 4, 6, 12): 2 odd and 4 even including adjusting factors (3, 4, and 12). a(n) = 2*2 + 3*4 - 3 - 1 - 1 + 1 = 12: (1) a 0-dimensional simplex with 12 coincident points; (2) a 1-dimensional simplex with 2 groups of 6 coincident points; (3) a 2-dimensional simplex with 3 groups of 4 coincident points; (4,5) a square and a 3-dimensional simplex each with 4 groups of 3 coincident points; (6,7,8) a hexagon, an octahedron, and a 5-dimensional simplex each with 2 coincident points at the vertices; (9, 10, 11, 12) a dodecagon, a 6-dimensional orthoplex, an 11-dimensional simplex, and an icosahedron each with no coincident points.
For n = 20, the set of factors of 20 is (1, 2, 4, 5, 10, 20): 2 odd and 4 even including adjusting factors (4 and 20). a(n) = 2*2 + 3*4 - 3 - 1 + 1 = 13: (1) a 0-dimensional simplex with 20 coincident points; (2) a 1-dimensional simplex with 2 groups of 10 coincident points; (3, 4) a square and a 3-dimensional simplex each with 4 groups of 5 coincident points; (5, 6) a pentagon, and a 4-dimensional simplex each with groups of 4 coincident points; (7, 8, 9) a decagon, a 5-dimensional orthoplex, and a 9-dimensional simplex each with 2 coincident points at the vertices; (10, 11, 12, 13) a 20-sided polygon, a 10-dimensional orthoplex, a 19-dimensional simplex, and a dodecahedron.
For n = 24, the set of factors of 24 is (1, 2, 3, 4, 6, 8, 12, 24): 2 odd and 6 even including adjusting factors (3, 4, 8, 12, and 24). a(n) = 2*2 + 3*6 - 3 - 1 - 1 + 1 + 1 + 1 = 20: (1) a 0-dimensional simplex with 24 coincident points; (2) a 1-dimensional simplex with 2 groups of 12 coincident points; (3) a 2-dimensional simplex with 3 groups of 8 coincident points; (4, 5) a square and a 3-dimensional simplex each with 4 groups of 6 coincident points; (6, 7, 8) a hexagon, an octahedron, and a 5-dimensional simplex each with 4 coincident points; (9, 10, 11, 12) an octagon, a cube, a 4-dimensional orthoplex, a 7-dimensional simplex each with 3 coincident points; (13, 14, 15, 16) a dodecagon, a 6-dimensional orthoplex, an 11-dimensional simplex, and an icosahedron each with 2 coincident points; (17, 18, 19, 20) a 24-sided polygon, a 4-dimensional 24-cell, a 12-dimensional orthoplex, and a 23-dimensional simplex.

E. W. Weisstein, CRC Encyclopedia of Mathematics, 3rd Ed., CRC Press, 2009, 3037-3038.

Cf. A053016, A063723, A063927, A111336.

a(n) = Sum_{i|n} A111336(i).

A243842 Pair deficit of the most equal partition of n into two parts using standard rounding of the expectations of n, floor(n/2) and n-floor(n/2), assuming equal likelihood of states defined by the number of 2-cycles.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1

Offset: 0

Rajan Murthy and Vale Murthy, Jun 12 2014

nonn

The expectation for n = 2 is 0.5 so this is the first and only integer n for which the convention of rounding a half to an even number is pertinent. This affects a(2), a(3), a(4), and a(5). For n>2, twice the expectation, 2*E(n) must be an odd integer for this situation to arise. 2*E(n) = n*(n-1)*I(n-2)/I(n) for n>=2, where I(n) = A000085(n).
First notice that gcd(I(n), I(n-2)) = gcd(I(n-1) + (n-1)*I(n-2), I(n-2)) = gcd(I(n-1), I(n-2)). Now, suppose that there is an odd prime factor s that divides both I(m-1) and I(m-2) for some m. This would then imply that I(m), I(m+1), I(m+2), ... would all be a multiple of s, i.e., I(n) mod s would be zero for all n greater than or equal to m. A result of Chowla implies that I(n) mod s equals 1 infinitely often for any fixed odd prime s. This is a contradiction of the initial supposition. In other words, there is no odd prime factor that divides both I(m-1) and I(m-2), hence no odd prime factor in common between I(m) and I(m-2).
We can rewrite 2*E(n) as n*(n-1)*(2^a)*p/((2^b)*q), where gcd(p,q) = 1 and both p and q are odd. Using the result from Kim regarding b as a function of n, it can be shown that q > 2^(n/2) > n*(n-1) for all n greater than or equal to 16. Since q is larger than n*(n-1) we can reduce n*(n-1)/q to r/q', where gcd(r,q') = 1, q' odd, and q' is greater than 1. Let Z be an odd prime factor of q'. Z is not a divisor of r, p, or 2^a. Since Z is prime this implies that Z is not a divisor of the product r*2^a*p. Now rewrite 2*E(n) = r*(2^a)*p/(q'*(2^b)) as 2*E(n) = r*(2^a)*p/(Z*q"*(2^b)), where Z is an odd prime such that q' = Z*q". Let us suppose that 2*E(n) is an integer then 2*E(n)*Z*q"*(2^b) = r*(2^a)*p. This implies that Z is a divisor of r*(2^a)*p. This is a contradiction. We conclude that 2*E(n) is not an integer for n greater than or equal to 16. The remaining cases for 2*E(n) between 2 and 16 can be verified numerically.
Interestingly, given the recurrence relation I(n) = I(n-1) + (n-1)*I(n-2), 2*E(n) = n - n*I(n-1)/I(n).  Defining J(n) as I(n)/I(n-1), this yields 2*E(n) = n - n/J(n) where J(n) = 1+(n-1)/J(n-1). n/J(n) happens to be the finite continued fraction n/1+ (n-1)/1+ ...3/1+ 2/(1+1).

			Trivially, for n = 0,1 no pairs are possible so a(0) and a(1) are 0.
For n = 2, the expectation, E(n), equals 0.5. So a(2) = round(E(2)) - round(E(1)) - round(E(1)) = 0.
For n = 5 = 2 + 3, E(5) = 20/13, E(2) = 0.5 and E(3) = 0.75 and a(5) = round(E(5)) - round(E(2)) - round(E(3)) = 1.

Oskar Perron, Die Lehre von den Kettenbrüchen Band I, II, B. G. Teubner, 1954.

S. Chowla, I. N. Hernstein, and W. K. Moore, On recursions connected with symmetric groups. I, Canadian Journal of Mathematics, 3 (1951), 328-334.
Dongsu Kim, and J.S. Kim, A combinatorial approach to the power of 2 in the number of involutions, arXiv:0902.4311 [math.CO], 2009-2010.
Dongsu Kim, and J.S. Kim, A combinatorial approach to the power of 2 in the number of involutions, Journal of Combinatorial Theory, Series A 117 (8) (2010): 1082-1094
Wikipedia, Generalized continued fraction
Wikipedia, Telephone number (mathematics)

Cf. A162970 (numerator for calculating the expected value).
Cf. A000085 (denominator for calculating the expected value).
Cf. A243840 (analogous using floor rounding).
Cf. A243841 (analogous using ceiling rounding).

Let Er(n) = round(A162970(n)/A000085(n)). Then a(n) = Er(n) - Er(floor(n/2)) - Er(n-floor(n/2)).

A243841 Pair deficit of the most nearly equal partition of n into two parts using ceiling rounding of the expectations of n, floor(n/2) and n-floor(n/2), assuming equal likelihood of states defined by the number of 2-cycles.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1

Offset: 0

Rajan Murthy, Jun 12 2014

A162970 and A000085 provide the numerator and the denominator for calculating the expected value.

			Trivially, for n = 0,1 no pairs are possible so a(0) and a(1) are 0.
For n = 2, the expectation, E(n), equals 0.5. So a(2) = ceiling(E(2)) - (ceiling(E(1)) + ceiling(E(1))) = 1.
For n = 5 = 2 + 3, E(5) = 20/13, E(2) = 0.5 and E(3) = 0.75 and a(5) = ceiling(E(5)) - (ceiling(E(2)) + ceiling(E(3))) = 0.
Interestingly, for n = 8, E(8) = 532/191 and E(4) = 6/5, so a(n) = 3 - (2 + 2) = -1.

Cf. A000085, A162970.

a(n) = ceiling(A162970(n)/A000085(n)) - (ceiling(A162970(floor(n/2))/A000085(floor(n/2))) + ceiling(A162970(n-floor(n/2))/A000085(n-floor(n/2)))).

A243840 Pair deficit of the most nearly equal in size partition of n into two parts using floor rounding of the expectations for n, floor(n/2) and n- floor(n/2), assuming equal likelihood of states defined by the number of two-cycles.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 2, 1, 2, 2, 2, 1, 2, 2, 2

Offset: 0

Rajan Murthy, Jun 12 2014

nonn

			Trivially, for n = 0,1 no pairs are possible so a(0) and a(1) are 0.
For n = 2, the expectation, E(n), equals 0.5. So a(2) = floor(E(2)) - floor(E(1)) - floor(E(1)) = 0.
For n = 5 = 2 + 3, E(5) = 20/13, E(2) = 0.5 and E(3) = 0.75 and a(5) = floor(E(5)) - floor(E(2)) - floor(E(3)) = 1.
Interestingly, for n = 8, E(8) = 532/191 and E(4) = 6/5, so a(n) = 2 - 1 - 1 = 0.

A162970 provides the numerator for calculating the expected value.

A000085 provides the denominator for calculating the expected value.

a(n) = floor(A162970(n)/A000085(n)) - floor(A162970(floor(n/2))/A000085(floor(n/2))) - floor(A162970(n-floor(n/2))/A000085(n-floor(n/2))).

A240692 Squared radii of circles which exactly encircle clusters in the A2 lattice in increasing order.

Original entry on oeis.org

0, 4, 9, 13, 21, 25, 28, 36, 39, 43, 49, 57, 63, 64, 67, 76, 81, 84, 91, 93, 97, 109, 111, 117, 121, 124, 129, 133, 144, 147, 148, 156, 157, 163, 171, 175, 183, 189, 193, 196, 199, 201

Offset: 0

Rajan Murthy, Apr 10 2014

nonn

			For n = 1, a(1) = 4, the squared distance to the corners furthest from the deep hole of three hexagons which share the deep hole as a corner.
For n= 3, a(3) = 13, the squared distance to the furthest corners of the 6 hexagons third most distant from the deep hole - which, when added to the 3 that are closest and the 3 that are second-closest, yields a total of 12, which is A038588(3).

Rajan Murthy, Generating code for the sequence (Scilab)

A038588(n) gives the corresponding cluster size starting at n = 1 (the first positive radius circle);

A000404 is the analog for a Cartesian lattice.

A240600 Number of partially filled hexagons in the first 120-degree circular sector of hexagonal lattice A_2 centered at deep hole along the edge of a circle also centered at the deep hole.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 3, 3, 3, 5, 4, 5, 5, 7, 5, 5, 5, 6, 6, 8, 6, 8, 7, 7, 7, 9, 7, 7, 7, 9, 8, 9, 9, 11, 9, 11, 9, 9, 9, 11, 9, 10, 10, 12, 10, 12, 12, 14, 12, 14, 13, 13, 11, 11, 11, 13, 13, 15, 13, 13, 13, 15, 14

Offset: 0

Rajan Murthy, Apr 09 2014

nonn

A(n) alternates between the numbers for circles which intersect points on the A2 lattice and the numbers for circles which pass in between the points on a lattice.

			for n = 1, the squared radius is in the open interval (0,1) and the corresponding circle passes through 1 hexagon.
for n = 14, the squared radius is 13 with the corresponding circle passing through the furthest corner of 2 hexagons and passing through 5 hexagons.

Rajan Murthy, Code for computing 3*A(n) as a function of R^2 (scilab)

A038588 gives the number of hexagons completely encircled in all three circular sectors.
Squared radii of alternate entries is given by the Loeschian numbers A003136.
A234300 is the analog for the 2-d Cartesian lattice.
A237708 is the analog for the 3-d Cartesian lattice.
A239353 is the analog for the 4-d Cartesian lattice.

A239355 Number of unit hypercubes, aligned with a four-dimensional Cartesian mesh, partially enclosed along the edge of the first 2^4-ant of a hypersphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

0, 1, 1, 5, 5, 11, 11, 15, 14, 19, 19, 31, 31, 43, 39, 43, 43, 49, 49, 65, 59, 77, 77, 89, 85, 93, 89, 105, 105, 129, 117, 129, 128, 133, 133, 157, 145, 175, 171, 187, 181, 199, 195, 223, 211, 235, 223, 235, 235, 247, 235, 263, 257, 299, 287, 315, 303, 315

Offset: 1

Rajan Murthy, Mar 16 2014

nonn

			At radius 0, there are no partially filled cubes.  At radius > 0 but < 1, there is 1 partially filled square along the edge of the sphere.  At radius = 1, there is 1 partially filled cube along the edge of the sphere.  At radius > 1 but < sqrt(2), there are 5 partially filled cubes.

Rajan Murthy, Table of n, a(n) for n = 1..201

Cf. A001477 (corresponds to the square radius of alternate entries).

Cf. A237708 (3-dimensional analog), A234300 (2-dimensional analog).

Terms a(22) and beyond from b-file by Andrew Howroyd, Feb 05 2018

A239353 Number of unit hypercubes, aligned with a four-dimensional Cartesian mesh, completely within the first 2^4-ant of a hypersphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

1, 5, 11, 15, 19, 31, 32, 44, 48, 54, 58, 70, 82, 94, 100, 112, 124, 148, 164, 176, 194, 206, 219, 235, 247, 275, 281, 317, 333, 345, 369, 393, 417, 421, 437

Offset: 1

Rajan Murthy, Mar 16 2014

nonn

			When the radius of the sphere reaches 2, one cube is completely within the sphere. When the radius reaches 7^(1/2), five cubes are completely within the sphere.

Rajan Murthy, Table of n, a(n) for n = 1..100
Rajan Murthy, Scilab code for generating the sequence

Cf. A237707 (3-dimensional analog), A232499 (2-dimensional analog). The square radii corresponding to the elements of {a(n)} are the indices of the nonzero terms of A025428.

A237708 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, partially encircled along the edge of the first octant of a sphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

0, 1, 1, 4, 4, 7, 6, 7, 7, 10, 10, 16, 13, 16, 16, 19, 16, 22, 22, 28, 25, 28, 27, 28, 28, 34, 28, 34, 34, 37, 34, 43, 40, 46, 43, 46, 46, 52, 46, 52, 49, 52, 49, 52, 52, 61, 55, 67, 63

Offset: 0

Rajan Murthy, Feb 11 2014

nonn

			At radius 0, there are no partially filled cubes.  At radius >0 but < sqrt(1), there is 1 partially filled square along the edge of the sphere.  At radius = sqrt(1), there is 1 partially filled cube along the edge of the sphere.  At radius > 1 but < sqrt(2), there  are 4 partially filled cubes.

Cf. A000378 (corresponds to the square radius of alternate entries).

Cf. A234300 (2-dimensional analog).

A237707 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

Original entry on oeis.org

1, 4, 7, 10, 11, 17, 20, 23, 26, 32, 35, 38, 44, 48, 54, 60, 66, 69, 75, 78, 87, 96, 102, 105, 108, 114, 120, 121, 127, 133, 139, 145, 157, 163, 169, 178, 184, 196, 202, 214, 217, 220, 232, 238, 241, 244, 256, 263, 266, 278, 284, 296, 299, 308, 314, 329, 332

Offset: 1

Rajan Murthy, Feb 11 2014

nonn

			When the radius of the sphere reaches 3^(1/2), one cube is completely within the sphere. When the radius reaches 6^(1/2), four cubes are completely within the sphere.

Rajan Murthy, Table of n, a(n) for n = 1..200
Rajan Murthy, Table of n, a(n), and squared radius for n = 1..200
Rajan Murthy, Scilab program for this sequence
Charles R Greathouse IV, Illustration of this sequence

The radii corresponding to the terms are given by the square roots of A000408 starting with squared radius 3.
Cf. A232499 (2-dimensional analog).
Partial sums of A014465 and A063691 (but then with repeated terms omitted).

(* Illustrates the sequence *)
Cube[x_,y_,z_]:=Cuboid[{x-1,y-1,z-1},{x,y,z}]
Cubes[r_]:=Cube@@#&/@Select[Flatten[Table[{x,y,z},{x,1,r},{y,1,r},{z,1,r}],2],Norm[#]<=r&]
Draw[r_]:=Graphics3D[Union[Cubes[r],{{Green, Opacity[0.3], Sphere[{0,0,0},r]}}],PlotRange->{{0,r},{0,r},{0,r}},ViewPoint->{r,3r/4,3r/5}];
Draw/@Sqrt/@{3,6,9,11,12,14} (* Charles R Greathouse IV, Mar 12 2014 *)

Scilab
```
// See Murthy link.
```

a(n) ~ (Pi*sqrt(30)/25)*n^(3/2). - Charles R Greathouse IV, Mar 14 2014

Duplicate terms deleted by Rajan Murthy, Mar 06 2014

Terms a(36) and beyond added from b-file by Andrew Howroyd, Feb 27 2018

cp's OEIS Frontend

User: Rajan Murthy

A352622 Number of regular convex polytopes that can be formed with n indistinguishable points located at the vertices, coinciding in equal frequency at each vertex, if coinciding at all.

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A243842 Pair deficit of the most equal partition of n into two parts using standard rounding of the expectations of n, floor(n/2) and n-floor(n/2), assuming equal likelihood of states defined by the number of 2-cycles.

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A243841 Pair deficit of the most nearly equal partition of n into two parts using ceiling rounding of the expectations of n, floor(n/2) and n-floor(n/2), assuming equal likelihood of states defined by the number of 2-cycles.

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A243840 Pair deficit of the most nearly equal in size partition of n into two parts using floor rounding of the expectations for n, floor(n/2) and n- floor(n/2), assuming equal likelihood of states defined by the number of two-cycles.

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A240692 Squared radii of circles which exactly encircle clusters in the A2 lattice in increasing order.

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A240600 Number of partially filled hexagons in the first 120-degree circular sector of hexagonal lattice A_2 centered at deep hole along the edge of a circle also centered at the deep hole.

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A239355 Number of unit hypercubes, aligned with a four-dimensional Cartesian mesh, partially enclosed along the edge of the first 2^4-ant of a hypersphere centered at the origin, ordered by increasing radius.

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A239353 Number of unit hypercubes, aligned with a four-dimensional Cartesian mesh, completely within the first 2^4-ant of a hypersphere centered at the origin, ordered by increasing radius.

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A237708 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, partially encircled along the edge of the first octant of a sphere centered at the origin, ordered by increasing radius.

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A237707 Number of unit cubes, aligned with a three-dimensional Cartesian mesh, completely within the first octant of a sphere centered at the origin, ordered by increasing radius.

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