A257594 -id:A257594 - cp's OEIS frontend

A182619 Number of vertices that are connected to two edges in a spiral without holes constructed with n hexagons.

6, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24

Offset: 1

Omar E. Pol, Dec 13 2010

The structure shows a hexagonal growth as in A182618.
a(n) is the number of vertices of the convex parts of the perimeter of the structure.
This sequence can be constructed geometrically in the following manner: Construct a gapless array of n equal circles with the rule of always choosing an arrangement with the maximum number of completely enclosed inner circles. Then, a(n) equals the number of circles required to create a kissing perimeter around the original array. Examples: a(1) = 6 because it takes 6 circles to create a kissing perimeter around 1 circle. a(7) = 12 because it takes 12 circles to create a kissing perimeter around 7 circles, which are arranged with 1 circle in center surrounded by 6 kissing circles. One could describe this as the "kissing numbers of kissing circles" sequence. - Peter Woodward, Apr 25 2015
a(n) is also the size of the smallest hexagonal polyomino that admits a hole of size n (Cf. A257594). - Luca Petrone, Feb 28 2017

			For n=1 there is 1 hexagon, so a(1)= 6 because there are 6 vertices that are connected to two edges.
For n=2 there are 2 connected hexagons, so a(2)= 8 because there are 8 vertices that are connected to two edges.
For n=3 there are 3 connected hexagons, so a(3)= 9 because there are 9 vertices that are connected to two edges.
If written as a triangle, begins:
6,
8,9,10,11,12,12,
13,14,14,15,15,16,16,17,17,18,18,18,
19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24,24

Row n has A008458(n-1) terms.

Cf. A182618, A257594.

A204259 Matrix given by f(i,j) = 1 + [(2i+j) mod 3], by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jan 14 2012

This data is used to specify the height of hexagonally packed cylinders in a triangle with open boundaries. Three cylinders that touch each other define a "triple" and water can be retained between these cylinders. A257594, A258445 and A259052 give a classification for such spaces. The links below ignore the inter-cylinder space retention and only consider the water retention above solid cylinders. - Craig Knecht, Jul 16 2015

			Northwest corner:
1 2 3 1 2 3 1 2
3 1 2 3 1 2 3 1
2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2
3 1 2 3 1 2 3 1
2 3 1 2 3 1 2 3

Craig Knecht, Water retention triple.
Craig Knecht, Row sums of numbers completely surrounded by larger numbers (water retention) in A204259 = A008611.
Wikipedia, Water retention on mathematical surfaces

Cf. A204260.

f[i_, j_] := 1 + Mod[2 i + j, 3];
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]   (* A204259 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 20}]     (* A204258 *)

A257481 Consider a hole-less cluster of n circles in the hexagonal lattice packing of circles; a(n) is the maximal number of circles that touch 6 circles.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4

Offset: 1

Peter Woodward, Apr 26 2015

			For a(7), one circle can be completely enclosed by six surrounding circles, so a(7)=1, a(n)=0 for n<7.
For a(10), two circles can be completely enclosed by eight surrounding circles, so a(10)=2.

R. L. Graham and N. J. A. Sloane, Penny-Packing and Two-Dimensional Codes, Discrete and Comput. Geom. 5 (1990), 1-11.
Kival Ngaokrajang, Illustration of initial terms
Eric Weisstein's World of Mathematics, Hexagonal grid
Wikipedia, Circle packing

Cf. A182619, A257594, A069813.

Edited by N. J. A. Sloane, May 18 2015

A260302 Maximum water retention of a number octagon of order n.

Original entry on oeis.org

0, 0, 231, 1378, 4753, 12246, 26335, 50086, 87153, 141778, 218791, 323610, 462241, 641278, 867903, 1149886, 1495585, 1913946, 2414503, 3007378, 3703281, 4513510, 5449951, 6525078, 7751953, 9144226, 10716135, 12482506, 14458753, 16660878, 19105471, 21809710

Offset: 1

Craig Knecht, Nov 10 2015

A number octagon fills an octagon on a square grid with the smallest unique natural numbers.

The sum of the interior values for a number hexagon on a circular lattice is A079903. There are nice illustrations for this by Mathar at A257594.

			      (22 23 24)
   (37  1  2  3 25)
(36  4  5  6  7  8  26)
(35  9 10 11 12 13  27)
(34 14 15 16 17 18  28)
   (33 19 20 21 29)
      (32 31 30)
The largest values (22 - 37) form the dam with the value 22 being the spillway.

G. C. Greubel and Craig Knecht, Table of n, a(n) for n = 1..1000 (Terms 1 through 32 were computed by Craig Knecht; terms 33 through 1000 by G. C. Greubel, Nov 13 2015; term 451 = 1002105368551 corrected by _Georg Fischer_, May 24 2019)
Craig Knecht, Octagon volume calculation.
Craig Knecht, Volume retention of a number square.
Wikipedia, Water retention on mathematical surfaces
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A261347 (water retention on a number square).

[0,0] cat [(1/2)*(7*n^2-18*n+12)*(7*n^2-18*n+13): n in [3..60]]; // Vincenzo Librandi, Nov 20 2015

Table[-KroneckerDelta[n,1]  - 10*KroneckerDelta[n,2] + (1/2)*((7*n^2-18*n+12)^2+(7*n^2-18*n+12)), {n, 1, 30}] (* G. C. Greubel, Nov 13 2015 *)

concat(vector(2), Vec(-x^3*(10*x^4-49*x^3+173*x^2+223*x+231)/(x-1)^5 + O(x^100))) \\ Colin Barker, Nov 11 2015

a(n) =  (1/2)*(7*n^2 - 18*n + 12) (7*n^2 - 18*n + 13) for n > 2.
From Colin Barker, Nov 11 2015: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.
G.f.: -x^3*(10*x^4-49*x^3+173*x^2+223*x+231) / (x-1)^5. (End)

A343262 a(n) is the number of edges of a regular polygon P with the property that packing n nonoverlapping equal circles inside P, arranged in a configuration with dihedral symmetry D_{2m} with m >= 3, maximizes the packing density.

Original entry on oeis.org

3, 4, 5, 3, 6, 7, 4, 3, 5, 6, 6, 7, 3, 4, 4, 6, 6, 4, 3

Offset: 3

Ya-Ping Lu, Apr 09 2021

Numbers of dihedral symmetries D_{2m} (m >= 3) that n nonoverlapping equal circles possess are given in A343005. The regular polygon is a circle for n=1 and a square for n=2. However, as the symmetry types, O(2) for one circle and D_{4} for two circles, are not D_{2m} with m >= 3, the index of the sequence starts at n = 3.

It can be shown that a(n) <= n and a(n) = k*m/2, where m is the order of a dihedral symmetry of n-circle packing configurations and k is a positive integer.

			For n=3, 3-circle configurations possess one dihedral symmetry D_{6}, or m = 3. Since a(n) must be <= 3 and also a multiple of m, a(n) = 3.
For n = 16, 16-circle configurations have 6 D_{2m} symmetries with m >= 3.
Packing densities are for
m = 16: Pi/(2+2*csc(Pi/8)) = 0.43474+,
m = 15: (8*Pi/15)/(1+csc(2*Pi/15)) = 0.48445+,
m =  8: 4*sqrt(2)*Pi/(1+sqrt(2)+sqrt(3)+sqrt(4-2*sqrt(2)))^2 = 0.65004+,
m =  5: (16*Pi/5)*(7-3*sqrt(5))/sqrt(10+2*sqrt(5)) = 0.77110+,
m =  4: Pi/4 = 0.78539+,
m =  3: 8*Pi/(12+13*sqrt(3)) = 0.72813+.
The highest packing density is achieved at m = 4, or a(16) = 4.
Symmetry type (S) of n-circle configuration giving the highest packing density and the corresponding number of edges (N) of the regular polygon and packing density are given below. The packing configurations are illustrated in the Links.
   n       S      N      Packing density
------  --------  --  -------------------------------------------------------------
   3      D_{6}   3   Pi/(2+4/sqrt(3))                                   = 0.72900+
  4,9,16  D_{8}   4   Pi/4                                               = 0.78539+
   5      D_{10}  5   Pi/(2+8/sqrt(10+2*sqrt(5)))                        = 0.76569+
   6      D_{6}   3   6*Pi/(12+7*sqrt(3))                                = 0.78134+
   7      D_{12}  6   7*Pi/(12+8*sqrt(3))                                = 0.85051+
   8      D_{14}  7   4*Pi/(7+7/sin(2*Pi/7))                             = 0.78769+
  10      D_{6}   3   5*Pi/(9+6*sqrt(3))                                 = 0.81001+
  11      D_{10}  5   (22*Pi/25)/sqrt(10+2*sqrt(5))                      = 0.72671+
  12      D_{6}   6   6*Pi/(12+7*sqrt(3))                                = 0.78134+
  13      D_{12}  6   13*sqrt(3)*Pi/96                                   = 0.73685+
  14      D_{14}  7   4*Pi/(sin(2*Pi/7)*(sqrt(3)+cot(Pi/7)+sec(Pi/7))^2) = 0.66440+
  15      D_{6}   3   15*Pi/(24+19*sqrt(3))                              = 0.82805+
  17      D_{8}   4   (17*Pi/4)/(7+3*sqrt(2)+3*sqrt(3)+sqrt(6))          = 0.70688+
  18      D_{12}  6   9*Pi/(12+13*sqrt(3))                               = 0.81915+
  19      D_{12}  6   19*Pi/(24+26*sqrt(3))                              = 0.86465+
  20      D_{8}   4   20*Pi/(2+sqrt(2)+2*sqrt(3)+sqrt(6))^2              = 0.72213+
  21      D_{6}   3   21*Pi/(30+28*sqrt(3))                              = 0.84045+

Erich Friedman, Packing Equal Copies
Ya-Ping Lu, Illustration of packing configurations
Eckard Specht, Packomania, Packings of equal and unequal circles in fixed-sized containers with maximum packing density

Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005.

A258186 a(n) is the maximal number of circles that can be enclosed by a closed chain of n circles; all circles' centers lie at coordinate points of the planar net 6.6.6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 6, 6, 7, 7, 10, 10, 13

Offset: 0

Kival Ngaokrajang, May 23 2015

Inspired by A257594, in which the centers of circles can be considered to be placed on coordinate points of the planar net 3.3.3.3.3.3. See illustrations in the links.

Kival Ngaokrajang, Illustration for n = 0..11, for n = 12..22

Cf. A257594.

A342843 a(n) is the number of edges of the regular polygon such that packing n nonoverlapping equal circles inside the regular polygon gives the highest packing density. a(n) = 0 if such a regular polygon is a circle.

Original entry on oeis.org

0, 4, 3, 4, 5, 3, 6, 7, 4, 3, 9, 6, 10, 6, 3, 4

Offset: 1

Ya-Ping Lu, Apr 12 2021

Terms for n = 11, 12, 13 and 14 are conjectured values supported by numerical results (see Packomania in the links).
It can be shown that a(n) <= n for n >= 3. As n increases, terms of values other than 3 and 6 will eventually disappear. For example, the packing density of triangular packing of more than 121 circles inside an equilateral triangle, or hexagonal packing of more than 552 circles inside a regular hexagon, is higher than that of square packing inside a square. Thus, for n > 121, the sequence does not have any terms with a(n) = 4.
Conjecture: As n tends to infinity, a(n) takes the value of 3 or 6 and the packing density approaches sqrt(3)*Pi/6.

			a(1) = 0. The maximum packing density for packing 1 circle in regular m-gon is (Pi/m)*cot(Pi/m), which is an increasing function of m. Highest packing density of 1 is achieved as m tends to infinity and the regular n-gon becomes a circle.
a(2) = 4. The maximum packing density for packing 2 circles in regular polygon with odd number of edges m >= 3 is 4*Pi/(m*sin(2*Pi/m))/(sec(Pi/(2*m))+sec(Pi/m))^2, which is smaller than the packing density in regular polygon with even number of edges m >= 4, 4*Pi/(m*sin(2*Pi/m))/(1+sec(Pi/m))^2, which is a decreasing function of m with a maximum of Pi/(3+2*sqrt(2)) at m = 4.
Symmetry type (S) of the n-circle configuration achieving the highest packing density and the corresponding number of edges (N) of the regular polygon and packing density for n up to 16 are listed below.
  n        S     N    Packing density
------  ------  ---   ---------------------------------------------------------
1       O(2)    oo                                                     1
2       D_{4}    4    Pi/(3+2*sqrt(2))                               = 0.53901+
3       D_{6}    3    (Pi/2)/(1+2/sqrt(3))                           = 0.72900+
4,9,16  D_{8}    4    Pi/4                                           = 0.78539+
5       D_{10}   5    (Pi/2)/(1+4/sqrt(10+2*sqrt(5)))                = 0.76569+
6       D_{6}    3    6*Pi/(12+7*sqrt(3))                            = 0.78134+
7       D_{12}   6    7*Pi/(12+8*sqrt(3))                            = 0.85051+
8       D_{14}   7    (4*Pi/7)/(1+1/sin(2*Pi/7))                     = 0.78769+
10      D_{6}    3    (5*Pi/3)/(3+2*sqrt(3))                         = 0.81001+
11      D_{2}    9    (11*Pi/18)/(1+csc(2*Pi/9))                     = 0.75120+
12      D_{6}    6    6*Pi/(12+7*sqrt(3))                            = 0.78134+
13      D_{2}   10    (13*Pi/20)/(1+sqrt(50+10*sqrt(5))/5)           = 0.75594+
14      D_{4}    6    (49*Pi/2)/(21+20*sqrt(3)+6*sqrt(7)+6*sqrt(21)) = 0.77737+
15      D_{6}    3    15*Pi/(24+19*sqrt(3))                          = 0.82805+

Erich Friedman, Packing Equal Copies
Eckard Specht, Packomania, Packings of equal and unequal circles in fixed-sized containers with maximum packing density

Cf. A023393, A051657, A084616, A084617, A084618, A084644, A133587, A227405, A247397, A253570, A257594, A269110, A308578, A337019, A337020, A343005, A343262.

A373612 Size of the largest polyiamond that can be enclosed in n cells on a triangular lattice.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 6, 6, 6, 7, 7, 10

Offset: 6

Abraham Maxfield, Jun 10 2024

			A single triangular face takes 12 triangles to completely enclose so a(12) = 1.

Cf. A290648. A257594 is the hexagonal tiling equivalent. A008642 is the square tiling equivalent (if prepended with 7 zeros).

cp's OEIS Frontend

A182619 Number of vertices that are connected to two edges in a spiral without holes constructed with n hexagons.

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A204259 Matrix given by f(i,j) = 1 + [(2i+j) mod 3], by antidiagonals.

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A257481 Consider a hole-less cluster of n circles in the hexagonal lattice packing of circles; a(n) is the maximal number of circles that touch 6 circles.

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A260302 Maximum water retention of a number octagon of order n.

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A343262 a(n) is the number of edges of a regular polygon P with the property that packing n nonoverlapping equal circles inside P, arranged in a configuration with dihedral symmetry D_{2m} with m >= 3, maximizes the packing density.

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A258186 a(n) is the maximal number of circles that can be enclosed by a closed chain of n circles; all circles' centers lie at coordinate points of the planar net 6.6.6.

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A342843 a(n) is the number of edges of the regular polygon such that packing n nonoverlapping equal circles inside the regular polygon gives the highest packing density. a(n) = 0 if such a regular polygon is a circle.

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A373612 Size of the largest polyiamond that can be enclosed in n cells on a triangular lattice.

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