A296368 -id:A296368 - cp's OEIS frontend

A298031 Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.

1, 4, 10, 16, 30, 36, 48, 54, 66, 72, 84, 90, 102, 108, 120, 126, 138, 144, 156, 162, 174, 180, 192, 198, 210, 216, 228, 234, 246, 252, 264, 270, 282, 288, 300, 306, 318, 324, 336, 342, 354, 360, 372, 378, 390, 396, 408, 414, 426, 432, 444, 450, 462, 468, 480, 486, 498, 504, 516, 522, 534, 540

Offset: 0

N. J. A. Sloane, Jan 21 2018; extended with formula, Jan 24 2018

Also known as the mta net.
This is one of the Laves tilings.
In the Ferreol link this is described as the dual to the Diana tiling. - N. J. A. Sloane, May 24 2020

Colin Barker, Table of n, a(n) for n = 0..1000
Robert Ferreol, Pavage de diane et son dual
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, The subgraph H shown in one quadrant of the graph of the tiling.
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A008574, A298032 (partial sums), A298029 (for a trivalent node), A298033 (hexavalent node).

List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.

f4:=proc(n) local L; L:=[1,4,10,16];
if n<4 then L[n+1] elif (n mod 2) = 0 then 9*n-6 else 9*n-9; fi;
end;
[seq(f4(n),n=0..80)];

Join[{1, 4, 10, 16}, LinearRecurrence[{1, 1, -1}, {30, 36, 48}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Vec((1 + 3*x + 5*x^2 + 3*x^3 + 8*x^4 - 2*x^6) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Theorem: For n >= 4, a(n) = 9*n-6 if n is even, otherwise a(n) = 9*n-9.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: -(2*x^6 - 8*x^4 - 3*x^3 - 5*x^2 - 3*x - 1) / ((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4. - Colin Barker, Jan 25 2018
a(n) = 6*A007494(n-1), n>3. - R. J. Mathar, Jan 29 2018

A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.

Original entry on oeis.org

1, 6, 12, 24, 30, 42, 48, 60, 66, 78, 84, 96, 102, 114, 120, 132, 138, 150, 156, 168, 174, 186, 192, 204, 210, 222, 228, 240, 246, 258, 264, 276, 282, 294, 300, 312, 318, 330, 336, 348, 354, 366, 372, 384, 390, 402, 408, 420, 426, 438, 444, 456, 462, 474, 480, 492, 498, 510, 516, 528, 534, 546, 552

Offset: 0

N. J. A. Sloane, Jan 21 2018, corrected Jan 24 2018

Also known as the mta net.

This is one of the Laves tilings.

Colin Barker, Table of n, a(n) for n = 0..1000
Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The mta tiling (or net)
N. J. A. Sloane, The Dual(3.4.6.4) tiling
N. J. A. Sloane, The subgraph H shown in one 60-degree sector of the graph of the tiling.
N. J. A. Sloane, Overview of coordination sequences of Laves tilings [Fig. 2.7.1 of Grünbaum-Shephard 1987 with A-numbers added and in some cases the name in the RCSR database]
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A008574, A038764 (partial sums), A298029 (coordination sequence for a trivalent node), A298031 (coordination sequence for a tetravalent node).

f6:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 9*n-6 else 9*n-3; fi; end;
[seq(f6(n),n=0..80)];

Join[{1}, LinearRecurrence[{1, 1, -1}, {6, 12, 24}, 62]] (* Jean-François Alcover, Apr 23 2018 *)

Vec((1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

apply( {A298033(n)=if(n,n*3\/2*6-6,1)}, [0..66]) \\ M. F. Hasler, Jan 11 2022

Theorem: For n>0, a(n) = 9*n-6 if n is even, a(n) = 9*n-3 if n is odd.
The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. The subgraph H is shown above in the links.
G.f.: (1 + 5*x + 5*x^2 + 7*x^3) / ((1 - x)*(1 - x^2)).
First differences are 1, 5, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, ...
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3. - Colin Barker, Jan 25 2018
a(n) = 6*floor((3n-1)/2) for n > 0. - M. F. Hasler, Jan 11 2022

A321019 Coordination sequence for 3-D tiling (Cairo tiling) X Z, with respect to a 5-valent point.

Original entry on oeis.org

1, 5, 16, 36, 63, 98, 143, 196, 255, 322, 399, 484, 575, 674, 783, 900, 1023, 1154, 1295, 1444, 1599, 1762, 1935, 2116, 2303, 2498, 2703, 2916, 3135, 3362, 3599, 3844, 4095, 4354, 4623, 4900, 5183, 5474, 5775, 6084, 6399, 6722, 7055, 7396, 7743, 8098, 8463, 8836, 9215, 9602, 9999

Offset: 0

N. J. A. Sloane, Nov 10 2018

By (Cairo tiling) X Z is meant a stack of layers of the planar Cairo tiling at integer levels.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A040000, A296368.

Vec((1 + x)*(1 + x + 4*x^2 + 2*x^4 + x^5 - x^6) / ((1 - x)^3*(1 + x^2)) + O(x^50)) \\ Colin Barker, Nov 11 2018

G.f.: (x^6-x^5-2*x^4-4*x^2-x-1)*(1+x)/((x^2+1)*(x-1)^3), which is the product of the g.f.'s for A296368 and A040000.
From Colin Barker, Nov 11 2018: (Start)
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>5.
a(n) = (-2 + (-i)^(1+n) + i^(1+n) + 8*n^2) / 2 for n>2, where i=sqrt(-1).
(End)

A332019 The number of cells added in the n-th generation of the following procedure: start by coloring any triangle on the snub square tiling, then repeatedly color every cell that shares a vertex with a colored cell.

Original entry on oeis.org

1, 9, 21, 35, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 588, 600, 612, 624, 636

Offset: 1

Peter Kagey, Feb 04 2020

Peter Kagey, Table of n, a(n) for n = 1..1000
Code Golf Stack Exchange, Concentric rings on a snub square tiling.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594.

A296368 is the analogous sequence when instead coloring every cell that shares a side with a colored cell.

a(n) = 12*(n - 1) for n > 4.
From Stefano Spezia, Feb 05 2020: (Start)
G.f.: x*(1 + 7*x + 4*x^2 + 2*x^3  - x^4 - x^5)/(-1 + x)^2.
a(n) = 2*a(n-1) - a(n-2) for n > 6.
(End)

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A298031 Coordination sequence of Dual(3.4.6.4) tiling with respect to a tetravalent node.

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A298033 Coordination sequence of the Dual(3.4.6.4) tiling with respect to a hexavalent node.

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A321019 Coordination sequence for 3-D tiling (Cairo tiling) X Z, with respect to a 5-valent point.

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A332019 The number of cells added in the n-th generation of the following procedure: start by coloring any triangle on the snub square tiling, then repeatedly color every cell that shares a vertex with a colored cell.

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