A005899 -id:A005899 - cp's OEIS frontend

A236967 Expansion of (1+3x)^2/(1-3x)^2.

1, 12, 72, 324, 1296, 4860, 17496, 61236, 209952, 708588, 2361960, 7794468, 25509168, 82904796, 267846264, 860934420, 2754990144, 8781531084, 27894275208, 88331871492, 278942752080, 878669669052, 2761533245592, 8661172452084, 27113235502176, 84728860944300

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 22 2014

Index entries for linear recurrences with constant coefficients, signature (6,-9)

Cf. Expansion of (1 + k*x)^m/(1 - k*x)^m where the values of k,m are:
......|..m = 1..|..m = 2..|..m = 3..|..m = 4..|..m = 5..|..m = 6..|
k = 1 | A040000 | A008574 | A005899 | A008412 | A008413 | A008414 |
k = 2 | A151821 | A241204 |         |         |         |         |
k = 3 | A099856 | A236967 |         |         |         |         |
k = 4 | A081654 |         |         |         |         |         |
-------------------------------------------------------------------

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+3*x)^2/(1-3*x)^2));

For n >= 1, a(n) = 4*n*3^n. - Robert Israel, May 08 2014

Edited by Wolfdieter Lang, May 07 2014

A242941 a(n) is the number of convex uniform tessellations in dimension n.

Original entry on oeis.org

1, 11, 28, 143

Offset: 1

Felix Fröhlich, May 27 2014

Terms for n > 4 have not been determined so far. Alfredo Andreini in 1905 gave a value of 25 for a(3), later found to be incorrect. The value 28 for a(3) was given by Norman Johnson in 1991 and later in 1994 independently by Branko Grünbaum. The value for a(4) was given by George Olshevsky in 2006.
Deza and Shtogrin (2000) agree that the value of a(3) is 28, although the authors do not provide a proof. - Felix Fröhlich, Nov 29 2014
From Felix Fröhlich, Feb 03 2019: (Start)
The 11 convex uniform tilings are all illustrated in Kepler, 1619. For an argument that exactly 11 such tilings exist, see Grünbaum, Shephard, 1977.
In dimension 2, the definition of "uniform polytope" usually seems to be equivalent to the regular polygons in order to exclude polygons that alternate two different edge-lengths. Applying this principle retroactively to dimension 1 (as done, as I assume, by Coxeter, see Coxeter, 1973, p. 129) yields a(1) = 1. (End)

H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486614809.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, Vol. 4, No. 2 (1994), 49-56.
N. W. Johnson, Uniform Polytopes, [To appear, cf. Weiss, Stehle, 2017].

A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle correspondenti reto correlative [On the regular and semiregular nets of polyhedra and on the corresponding correlative nets], Mem. Società Italiana della Scienze, Ser.3, 14 (1905), 75-129.
M. Deza and M. Shtogrin, Uniform Partitions of 3-space, their Relatives and Embedding, European Journal of Combinatorics, Vol. 21, No. 6 (2000), 807-814.
B. Grünbaum and G. C. Shephard, Tilings by Regular Polygons, Mathematics Magazine, Vol. 50, No. 5 (1977), 227-247.
J. Kepler, Harmonices Mundi [The Harmony of the World] (1619).
G. Olshevsky, Uniform Panoploid Tetracombs (2006)
A. I. Weiss and E. M. Stehle, Norman W. Johnson (12 November 1930 to 13 July 2017), The Art of Discrete and Applied Mathematics, Vol. 1, No. 1 (2018).
Wikipedia, Convex uniform honeycomb
Wikipedia, List of convex uniform tilings

Cf. A068599.
List of coordination sequences for the 11 uniform 2D tilings: A008458(the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706(3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for the 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

Edited by N. J. A. Sloane, Feb 15 2018

Edited by Felix Fröhlich, Feb 03-10 2019

A346958 a(n) is the minimal number of cubes required to make a void of volume n.

Original entry on oeis.org

6, 10, 13, 15, 17, 18, 18, 21, 23, 25, 26, 26

Offset: 1

Mohammed Yaseen, Aug 08 2021

Following is an illustration of the first few voids in the form of polycubes (where an o represents a continuation upwards and an x represents a continuation downwards) each of which can be made by concealing it with a(n) cubes.
                          .---.       .---.
                          |   |       |   |
  .---.   .---.---.   .---.---.   .---.---.
  |   |   |   |   |   |   |   |   |   | o |
  .---.   .---.---.   .---.---.   .---.---.
   n=1       n=2         n=3         n=4
      .---.       .---.           .---.
      |   |       |   |           |   |
  .---.---.   .---.---.---.   .---.---.---.
  |   | o |   |   | o |   |   |   | ox|   |
  .---.---.   .---.---.---.   .---.---.---.
      |   |       |   |           |   |
      .---.       .---.           .---.
     n=5           n=6             n=7
Equivalently, the minimum perimeter size of any polycube of size n. - Sean A. Irvine, Aug 23 2021
Conjecture: When n is in A001845 the void is an octahedral crystal ball of volume n = A001845(m), which is concealed by a(n) = A005899(m+1) cubes. So a(A001845(m)) = A005899(m+1), m>=0. For example, a(1)=6 and a(7)=18. - Mohammed Yaseen, Sep 15 2022

			A cube-shaped void can be made by concealing it with 6 cubes, which is the minimal number to do so. So a(1)=6.
A dicube-shaped void can be made by concealing it with 10 cubes, which is the minimal number to do so. So a(2)=10.

Sean A. Irvine, Java program (github)

Cf. A261491 (2D analog).

Cf. A000162, A003211, A193416.

a(n) < A193416(n) for n>2.

a(8)-a(12) from Sean A. Irvine, Aug 23 2021

A350453 Number of Latin squares of order 2n with maximum inner distance with fixed entry 1 in cell (1,1).

Original entry on oeis.org

1, 144, 112, 340, 696, 1468, 2528, 4388, 6760, 10444, 14928, 21364, 28952, 39260, 51136, 66628, 84168, 106348, 131120, 161684, 195448, 236284, 280992, 334180, 391976, 459788, 533008, 617908, 709080, 813724, 925568, 1052804, 1188232, 1341100, 1503216, 1684948

Offset: 1

Omar Aceval Garcia, Dec 31 2021

The inner distance of a matrix with entries in [1,n] is the minimum of distances between vertically or horizontally adjacent entries. For example, every Latin square of order 2, 3, or 4 has inner distance 1, since there are consecutive integers which are adjacent. The distance between x and y in [1,n] with x < y is the minimum of y - x and n + x - y.

			For example there are 144 Latin squares of order 4 (with a 1 in the top left), all of which have maximum inner distance. There are only 112 such Latin squares of order 6, 340 of order 8, etc.
Every Latin square of order 4 by default has the maximum inner distance; the same is not true for any order higher than 4, which may explain why a(2) > a(3).

Jinyuan Wang, Table of n, a(n) for n = 1..1000
Omar Aceval Garcia, On the Number of Maximum Inner Distance Latin Squares, arXiv:2112.13912 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A005899, A069894, A248800.

a(n) = 4*n + ( n^2 + 3/2 + (1/2)*(-1)^n )^2 for n >= 3.
a(n) = 4*n + A248800(n)^2 for n >= 3.
For n >= 5, a(n) - a(n-2) = 8*n^3 - 24*n^2 + (44 + 4*(-1)^n)*n - 20 - 4*(-1)^n.
For n >= 7, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) + (48 + 16*(-1)^n)*(n-2).
G.f.: x*(1 + 142*x - 178*x^2 - 166*x^3 + 656*x^4 + 62*x^5 - 622*x^6 + 190*x^7 + 207*x^8 - 100*x^9)/((1 - x)^5*(1 + x)). - Stefano Spezia, Jan 01 2022

More terms from Jinyuan Wang, Jan 01 2022

A354961 Coordination sequence for the Manhattan cubic lattice.

Original entry on oeis.org

1, 3, 9, 21, 42, 67, 109, 145, 207, 255, 337, 397, 499, 571, 693, 777, 919, 1015, 1177, 1285, 1467, 1587, 1789, 1921, 2143, 2287, 2529, 2685, 2947, 3115, 3397, 3577, 3879, 4071, 4393, 4597, 4939, 5155, 5517, 5745, 6127, 6367, 6769, 7021, 7443, 7707, 8149, 8425

Offset: 0

Rémy Sigrist, Jun 13 2022

nonn

In the Manhattan cubic lattice, streets parallel to one of the 3 axis have alternating orientations:
                      .---->----.
                     /|        /|
                    v |       ^ |
                   /  v      /  ^
                  .----<----.   |
                  |   |     |   |
                  |   .----<|---.
                  ^  /      v  /
                  | ^       | v
                  |/        |/
                  O---->----.

Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, C# program
Index entries for coordination sequences

Cf. A005899, A336627 A354957.

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A236967 Expansion of (1+3*x)^2/(1-3*x)^2.

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A242941 a(n) is the number of convex uniform tessellations in dimension n.

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A346958 a(n) is the minimal number of cubes required to make a void of volume n.

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A350453 Number of Latin squares of order 2n with maximum inner distance with fixed entry 1 in cell (1,1).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A354961 Coordination sequence for the Manhattan cubic lattice.

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Author

Keywords

Comments

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Crossrefs

A236967 Expansion of (1+3x)^2/(1-3x)^2.