A298036 -id:A298036 - cp's OEIS frontend

A298040 Coordination sequence of Dual(4.6.12) tiling with respect to a tetravalent node.

1, 4, 20, 24, 40, 40, 60, 56, 80, 72, 100, 88, 120, 104, 140, 120, 160, 136, 180, 152, 200, 168, 220, 184, 240, 200, 260, 216, 280, 232, 300, 248, 320, 264, 340, 280, 360, 296, 380, 312, 400, 328, 420, 344, 440, 360, 460, 376, 480, 392, 500, 408, 520, 424, 540

Offset: 0

N. J. A. Sloane, Jan 22 2018

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Tom Karzes, Tiling Coordination Sequences
N. J. A. Sloane, Illustration of initial terms (shows one 90-degree quadrant of 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 (0,2,0,-1).

Cf. A072154, A298041 (partial sums), A298036 (12-valent node), A298038 (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.

LinearRecurrence[{0,2,0,-1},{1,4,20,24,40,40},60] (* Harvey P. Dale, Apr 06 2022 *)

Conjecture: For n>0, a(n)=10n if n even, otherwise 8n.
Conjectures from Colin Barker, Apr 01 2020: (Start)
G.f.: (1 + 4*x + 18*x^2 + 16*x^3 + x^4 - 4*x^5) / ((1 - x)^2*(1 + x)^2).
a(n) = (9 + (-1)^n)*n for n>1.
a(n) = 2*a(n-2) - a(n-4) for n>5.
(End)

Terms a(8)-a(54) added by Tom Karzes, Apr 01 2020

A234275 Expansion of (1+2x+9x^2-4*x^3)/(1-x)^2.

Original entry on oeis.org

1, 4, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432

Offset: 0

N. J. A. Sloane, Dec 24 2013

Also the coordination sequence for a point of degree 4 in the tiling of the Euclidean plane by right triangles (with angles Pi/2, Pi/4, Pi/4). These triangles are fundamental regions for the Coxeter group (2,4,4). In the notation of Conway et al. 2008 this is the tiling *442. The coordination sequence for a point of degree 8 is given by A022144. - N. J. A. Sloane, Dec 28 2015

First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 899", based on the 5-celled von Neumann neighborhood. Initialized with a single black (ON) cell at stage zero. - Robert Price, May 28 2016

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5. See p. 191.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Colin Barker, Table of n, a(n) for n = 0..1000
J. Choi, N. Pippenger, Counting the Angels and Devils in Escher's Circle Limit IV, arXiv preprint arXiv:1310.1357, 2013.
Tom Karzes, Tiling Coordination Sequences
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
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]
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A022144, A008590.
For partial sums see A265056.
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.

Join[{1, 4}, LinearRecurrence[{2, -1}, {16, 24}, 60]] (* Jean-François Alcover, Jan 08 2019 *)

Vec(-(4*x^3-9*x^2-2*x-1)/(x-1)^2 + O(x^100)) \\ Colin Barker, Jul 10 2015

a(n) = A022144(n), n>1. - R. J. Mathar, Jan 11 2014
From Colin Barker, Jul 10 2015: (Start)
a(n) = 8*n, n>1.
a(n) = 2*a(n-1) - a(n-2) for n>3.
G.f.: -(4*x^3-9*x^2-2*x-1) / (x-1)^2.
(End)

A298014 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-long.

Original entry on oeis.org

1, 3, 9, 15, 18, 27, 37, 37, 44, 57, 54, 61, 77, 71, 78, 97, 88, 95, 117, 105, 112, 137, 122, 129, 157, 139, 146, 177, 156, 163, 197, 173, 180, 217, 190, 197, 237, 207, 214, 257, 224, 231, 277, 241, 248, 297, 258, 265, 317, 275, 282, 337, 292, 299, 357, 309, 316, 377, 326, 333, 397, 343, 350, 417, 360

Offset: 0

Chaim Goodman-Strauss and N. J. A. Sloane, Jan 11 2018

The snub-632 tiling in also called the fsz-d net. It is the dual of the 3.3.3.3.6 Archimedean tiling.

This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - Felix Fröhlich, Jan 13 2018

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.

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.
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 (0,0,2,0,0,-1).

Cf. A298015, A298016.

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.

f:=proc(n) local k,r,L; L:=[1,3,9,15,18];
if n<5 then L[n+1]
else k:=floor(n/3); r:=n-3*k;
  if r=0 then 20*k-3 elif r=1 then 17*k+3 else 17*k+10; fi;
fi; end;
[seq(f(n),n=0..80)];

Join[{1, 3, 9, 15, 18}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {27, 37, 37, 44, 57, 54}, 60]] (* Jean-François Alcover, Apr 28 2018 *)

Vec(-(x+1)*(2*x^9+x^7-5*x^6-3*x^5-6*x^4-6*x^3-7*x^2-2*x-1)/((x-1)^2*(x^2+x+1)^2) + O(x^60)) \\ Colin Barker, Jan 13 2018

For n >= 5, let k=floor(n/3). Then a(3*k) = 20*k-3, a(3*k+1)=17*k+3, a(3*k+2)=17*k+10.
G.f.: -(x+1)*(2*x^9+x^7-5*x^6-3*x^5-6*x^4-6*x^3-7*x^2-2*x-1)/((x-1)^2*(x^2+x+1)^2).
a(n) = 2*a(n-3) - a(n-6) for n>10. - Colin Barker, Jan 13 2018

A298015 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-short.

Original entry on oeis.org

1, 3, 6, 15, 21, 18, 33, 48, 30, 51, 72, 42, 69, 96, 54, 87, 120, 66, 105, 144, 78, 123, 168, 90, 141, 192, 102, 159, 216, 114, 177, 240, 126, 195, 264, 138, 213, 288, 150, 231, 312, 162, 249, 336, 174, 267, 360, 186, 285, 384, 198, 303, 408, 210, 321, 432, 222, 339, 456, 234, 357, 480, 246, 375

Offset: 0

Chaim Goodman-Strauss and N. J. A. Sloane, Jan 11 2018

The snub-632 tiling in also called the fsz-d net. It is the dual of the 3.3.3.3.6 Archimedean tiling.

This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - Felix Fröhlich, Jan 13 2018

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.

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. [Warning: there is an error in Eq. 8(b), the a(4) term should be changed from 24 to 21. With that correction Theorem then still holds. - _N. J. A. Sloane_, Apr 01 2020]
Tom Karzes, Illustration of a(0) to a(4) [Key: n, a(n), color: 0, 1, green; 1, 3, red; 2, 6, blue; 3, 15, purple; 4, 21, beige.]
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 (0,0,2,0,0,-1).

Cf. A298014, A298016.

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.

For n >= 6, let k=floor(n/3), so k >= 2. Then a(3*k) = 18*k-3, a(3*k+1)=24*k, a(3*k+2)=12*k+6. [Corrected by N. J. A. Sloane, Apr 01 2020]
a(n) = 2*a(n-3) - a(n-6) for n>=11. [Corrected by N. J. A. Sloane, Apr 01 2020]
G.f.: -(3*x^10-9*x^7-4*x^6-6*x^5-15*x^4-13*x^3-6*x^2-3*x-1)/(x^6-2*x^3+1). [Corrected by N. J. A. Sloane, Apr 01 2020]

a(4) corrected by Tom Karzes. I corrected the b-file and the formulas and deleted the programs. - N. J. A. Sloane, Apr 01 2020

A298016 Coordination sequence of snub-632 tiling with respect to a hexavalent node.

Original entry on oeis.org

1, 6, 12, 12, 24, 36, 24, 42, 60, 36, 60, 84, 48, 78, 108, 60, 96, 132, 72, 114, 156, 84, 132, 180, 96, 150, 204, 108, 168, 228, 120, 186, 252, 132, 204, 276, 144, 222, 300, 156, 240, 324, 168, 258, 348, 180, 276, 372, 192, 294, 396, 204, 312, 420, 216, 330, 444, 228, 348, 468, 240

Offset: 0

Chaim Goodman-Strauss and N. J. A. Sloane, Jan 11 2018

The snub-632 tiling in also called the fsz-d net. It is the dual of the 3.3.3.3.6 Archimedean tiling.

This is also called the "6-fold pentille" tiling in Conway, Burgiel, Goodman-Strauss, 2008, p. 288. - Felix Fröhlich, Jan 13 2018

J. H. Conway, H. Burgiel and Chaim Goodman-Strauss, The Symmetries of Things, A K Peters, Ltd., 2008, ISBN 978-1-56881-220-5.

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.
Chaim Goodman-Strauss and N. J. A. Sloane, Trunks and branches structure for calculating this sequence
Tom Karzes, Tiling Coordination Sequences
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 (0,0,2,0,0,-1).

Cf. A298014, A298015.

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.

f:=proc(n) local k,r;
if n=0 then return(1); fi;
r:=(n mod 3); k:=(n-r)/3;
if r=0 then 12*k elif r=1 then 18*k+6 else 24*k+12; fi;
end;
[seq(f(n),n=0..80)];

Join[{1}, LinearRecurrence[{0, 0, 2, 0, 0, -1}, {6, 12, 12, 24, 36, 24}, 60]] (* Jean-François Alcover, Apr 23 2018 *)

Vec((1 + 6*x + 12*x^2 + 10*x^3 + 12*x^4 + 12*x^5 + x^6) / ((1 - x)^2*(1 + x + x^2)^2) + O(x^60)) \\ Colin Barker, Jan 13 2018

For n >= 1, let k=floor(n/3). Then a(3*k) = 12*k, a(3*k+1)=18*k+6, a(3*k+2)=24*k+12.
a(n) = 2*a(n-3) - a(n-6) for n >= 7.
G.f.: -(-x^6-12*x^5-12*x^4-10*x^3-12*x^2-6*x-1)/(x^6-2*x^3+1).

A298022 Coordination sequence for Dual(3^3.4^2) tiling with respect to a trivalent node.

Original entry on oeis.org

1, 3, 7, 12, 17, 23, 28, 33, 37, 42, 47, 51, 56, 61, 65, 70, 75, 79, 84, 89, 93, 98, 103, 107, 112, 117, 121, 126, 131, 135, 140, 145, 149, 154, 159, 163, 168, 173, 177, 182, 187, 191, 196, 201, 205, 210, 215, 219, 224, 229, 233, 238, 243, 247, 252, 257, 261

Offset: 0

N. J. A. Sloane, Jan 21 2018

nonn

This tiling is also called the prismatic pentagonal tiling, or the cem-d net. It is one of the 11 Laves tilings.

B. Gruenbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. See p. 96.

Rémy Sigrist, Table of n, a(n) for n = 0..1000
Chung, Ping Ngai, Miguel A. Fernandez, Yifei Li, Michael Mara, Frank Morgan, Isamar Rosa Plata, Niralee Shah, Luis Sordo Vieira, and Elena Wikner. Isoperimetric Pentagonal Tilings, Notices of the AMS 59, no. 5 (2012), pp. 632-640. See Fig. 1 (right).
Tom Karzes, Tiling Coordination Sequences
Frank Morgan, Optimal Pentagonal Tilings, Video, May 2021. [Mentions this tiling]
Reticular Chemistry Structure Resource (RCSR), The cem-d tiling (or net)
Rémy Sigrist, PARI program for A298022
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]
N. J. A. Sloane, Illustration of initial terms

See A298023 for partial sums, A298024 for a tetravalent point.
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.
Cf. A049347.

PARI
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Conjectures from Colin Barker, Jan 22 2018: (Start)
G.f.: (1 + 2*x + 4*x^2 + 4*x^3 + 3*x^4 + 2*x^5 - 2*x^8) / ((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>5. (End)
Conjecture: a(n) = 2*(21*n + 3*A049347(n+2)/2)/9 for n > 4. - Stefano Spezia, Nov 24 2024

More terms from Rémy Sigrist, Jan 21 2018

A298026 Coordination sequence of Dual(3.6.3.6) tiling with respect to a hexavalent node.

Original entry on oeis.org

1, 6, 6, 18, 12, 30, 18, 42, 24, 54, 30, 66, 36, 78, 42, 90, 48, 102, 54, 114, 60, 126, 66, 138, 72, 150, 78, 162, 84, 174, 90, 186, 96, 198, 102, 210, 108, 222, 114, 234, 120, 246, 126, 258, 132, 270, 138, 282, 144, 294, 150, 306, 156, 318, 162, 330, 168, 342, 174, 354, 180, 366, 186, 378, 192, 390

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the kgd net.

This is one of the Laves tilings.

Robert Israel, Table of n, a(n) for n = 0..10000
Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
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 (0,2,0,-1).

Cf. A008579, A298027 (partial sums), A298028 (trivalent point).

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.

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

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {6, 6, 18, 12}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

Vec((1 + 6*x + 4*x^2 + 6*x^3 + x^4) / ((1 - x)^2*(1 + x)^2) + O(x^60)) \\ Colin Barker, Jan 22 2018

a(0)=1; a(2*k)=6*k, a(2*k+1)=12*k+6.
G.f.: 1 + 6*x*(1+x+x^2)/(1-x^2)^2. - Robert Israel, Jan 21 2018
From Colin Barker, Jan 22 2018: (Start)
a(n) = 3*n for n>0 and even.
a(n) = 6*n for n odd.
a(n) = 2*a(n-2) - a(n-4) for n>4.
(End)
a(n) = 6*A026741(n), n>0. - R. J. Mathar, Jan 29 2018

A298028 Coordination sequence of Dual(3.6.3.6) tiling with respect to a trivalent node.

Original entry on oeis.org

1, 3, 12, 9, 24, 15, 36, 21, 48, 27, 60, 33, 72, 39, 84, 45, 96, 51, 108, 57, 120, 63, 132, 69, 144, 75, 156, 81, 168, 87, 180, 93, 192, 99, 204, 105, 216, 111, 228, 117, 240, 123, 252, 129, 264, 135, 276, 141, 288, 147, 300, 153, 312, 159, 324, 165, 336, 171, 348, 177, 360, 183, 372, 189, 384, 195

Offset: 0

N. J. A. Sloane, Jan 21 2018

Also known as the kgd net.

This is one of the Laves tilings.

Robert Israel, Table of n, a(n) for n = 0..10000
Tom Karzes, Tiling Coordination Sequences
Reticular Chemistry Structure Resource (RCSR), The kgd tiling (or net)
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 (0,2,0,-1).

Cf. A008579, A135556 (partial sums), A298026 (trivalent point).
If the initial 1 is changed to 0 we get A165988 (but we need both sequences, just as we have both A008574 and A008586).
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.

f3:=proc(n) if n=0 then 1 elif (n mod 2) = 0 then 6*n else 3*n; fi; end;
[seq(f3(n),n=0..80)];

Join[{1}, LinearRecurrence[{0, 2, 0, -1}, {3, 12, 9, 24}, 80]] (* Jean-François Alcover, Mar 23 2020 *)

a(0)=1; a(2*k) = 12*k, a(2*k+1) = 6*k+3.
G.f.: 1 + 3*x*(x^2+4*x+1)/(1-x^2)^2. - Robert Israel, Jan 21 2018
a(n) = 3*A022998(n), n>0. - R. J. Mathar, Jan 29 2018

A298029 Coordination sequence of Dual(3.4.6.4) tiling with respect to a trivalent node.

Original entry on oeis.org

1, 3, 6, 12, 18, 33, 39, 51, 57, 69, 75, 87, 93, 105, 111, 123, 129, 141, 147, 159, 165, 177, 183, 195, 201, 213, 219, 231, 237, 249, 255, 267, 273, 285, 291, 303, 309, 321, 327, 339, 345, 357, 363, 375, 381, 393, 399, 411, 417, 429, 435, 447, 453, 465, 471, 483, 489, 501, 507, 519, 525, 537, 543, 555

Offset: 0

N. J. A. Sloane, Jan 21 2018

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

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, 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]
Wikipedia, Laves tilings
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A007310, A008574, A298030 (partial sums), A298031 (for a tetravalent node), A298033 (hexavalent node), A306771.

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.

Join[{1, 3, 6, 12, 18}, LinearRecurrence[{1, 1, -1}, {33, 39, 51}, 60]] (* Jean-François Alcover, Jan 07 2019 *)
Join[{1,3,6,12,18},Table[If[EvenQ[n],9n-15,9n-12],{n,5,70}]] (* Harvey P. Dale, Aug 25 2019 *)

Vec((1 + 2*x + 2*x^2 + 4*x^3 + 3*x^4 + 9*x^5 - 3*x^7) / ((1 - x)^2*(1 + x)) + O(x^60)) \\ Colin Barker, Jan 25 2018

Theorem: For n >= 5, if n is even then a(n) = 9*n-15, otherwise a(n) = 9*n-12. The proof uses the "coloring book" method described in the Goodman-Strauss & Sloane article. - N. J. A. Sloane, Jan 24 2018
G.f.: -(3*x^7 - 9*x^5 - 3*x^4 - 4*x^3 - 2*x^2 - 2*x - 1)/((1 - x)*(1 - x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>7. - Colin Barker, Jan 25 2018
a(n) = (3/2)*(6*n - (-1)^n - 9) for n>4. - Bruno Berselli, Jan 25 2018
a(n) = 3*A007310(n-1), n>4. - R. J. Mathar, Jan 29 2018

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

Original entry on oeis.org

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

cp's OEIS Frontend

A298040 Coordination sequence of Dual(4.6.12) tiling with respect to a tetravalent node.

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A234275 Expansion of (1+2*x+9*x^2-4*x^3)/(1-x)^2.

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A298014 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-long.

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A298015 Coordination sequence of snub-632 tiling with respect to a trivalent node of type short-short-short.

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A298016 Coordination sequence of snub-632 tiling with respect to a hexavalent node.

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A298022 Coordination sequence for Dual(3^3.4^2) tiling with respect to a trivalent node.

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A298026 Coordination sequence of Dual(3.6.3.6) tiling with respect to a hexavalent node.

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A234275 Expansion of (1+2x+9x^2-4*x^3)/(1-x)^2.