Displaying 1-10 of 37 results found.
1, 7, 19, 37, 55, 85, 121, 157, 205, 253, 307, 373, 439, 511, 589, 673, 763, 859, 961, 1063, 1177, 1297, 1417, 1549, 1681, 1819, 1969, 2119, 2275, 2437, 2605, 2779, 2959, 3145, 3331, 3529, 3733, 3937, 4153, 4369, 4591, 4825, 5059, 5299, 5545, 5797, 6055, 6319, 6589, 6859, 7141, 7429, 7717, 8017, 8317, 8623
COMMENTS
Linear recurrence and g.f. confirmed by Shutov/Maleev link in A301716. - Ray Chandler, Aug 30 2023
FORMULA
G.f.: (1 + 6*x + 12*x^2 + 17*x^3 + 12*x^4 + 17*x^5 + 12*x^6 + 6*x^7 + x^8) / ((1 - x)^3*(1 + x + x^2)*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) - a(n-8) + a(n-9) for n>8.
(End)
MATHEMATICA
Accumulate[LinearRecurrence[{0, 0, 1, 0, 1, 0, 0, -1}, {1, 6, 12, 18, 18, 30, 36, 36, 48}, 60]] (* or *) LinearRecurrence[{1, 0, 1, -1, 1, -1, 0, -1, 1}, {1, 7, 19, 37, 55, 85, 121, 157, 205}, 60] (* Harvey P. Dale, Sep 07 2024 *)
a(0) = 1, thereafter a(n) = 4n.
+10
123
1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 200, 204, 208, 212, 216, 220, 224, 228, 232
COMMENTS
Number of squares on the perimeter of an (n+1) X (n+1) board. - Jon Perry, Jul 27 2003 Coordination sequence for square lattice (or equivalently the planar net 4.4.4.4).
Apparently also the coordination sequence for the planar net 3.4.6.4. - Darrah Chavey, Nov 23 2014 I confirm that this is indeed the coordination sequence for the planar net 3.4.6.4. The points at graph distance n from a fixed point in this net essentially lie on a hexagon (see illustration in link).
If n = 3k, k >= 1, there are 2k + 1 nodes on each edge of the hexagon. This counts the corners of the hexagon twice, so the number of points in the shell is 6(2k + 1) - 6 = 4n. If n = 3k + 1, the numbers of points on the six edges of the hexagon are 2k + 2 (4 times) and 2k + 1 (twice), for a total of 12k + 10 - 6 = 4n. If n = 3k + 2 the numbers are 2k + 2 (4 times) and 2k + 3 twice, and again we get 4n points.
The illustration shows shells 0 through 12, as well as the hexagons formed by shells 9 (green, 36 points), 10 (black, 40 points), 11 (red, 44 points), and 12 (blue, 48 points).
It is clear from the net that this period-3 structure continues forever, and establishes the theorem.
In contrast, for the 4.4.4.4 planar net, the successive shells are diamonds instead of hexagons, and again the n-th shell (n > 0) contains 4n points.
Of course the two nets are very different, since 4.4.4.4 has the symmetry of the square, while 3.4.6.4 has only mirror symmetry (with respect to a point), and has the symmetry of a regular hexagon with respect to the center of any of the 12-gons. (End)
Also the coordination sequence for a 6.6.6.6 point in the 3-transitive tiling {4.6.6, 6.6.6, 6.6.6.6}, see A265045, A265046. - N. J. A. Sloane, Dec 27 2015 Also the coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_4].
Susceptibility series H_1 for 2-dimensional Ising model (divided by 2).
Also the Engel expansion of exp^(1/4); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002 Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00,0), (00;1) and (10;1). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1 < i2 and j1 < j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004 Also the coordination sequence for the htb net. - N. J. A. Sloane, Mar 31 2018 This is almost certainly also the coordination sequence for Dual(3.3.4.3.4) with respect to a tetravalent node. - Tom Karzes, Apr 01 2020 Minimal number of segments (equivalently, corners) in a rook circuit of a 2n X 2n board (maximal number is A085622). - Ruediger Jehn, Jan 02 2021
LINKS
Reticular Chemistry Structure Resource, sql and htbN. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
FORMULA
Euler transform of length 2 sequence [4, -2]. - Michael Somos, Apr 16 2007 G.f.: ((1 + x) / (1 - x))^2. E.g.f.: 1 + 4*x*exp(x). - Michael Somos, Apr 16 2007
EXAMPLE
Illustration of initial terms as perimeters of squares (cf. Perry's comment above):
. o o o o o o
. o o o o o o o
. o o o o o o o o
. o o o o o o o o o
. o o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
.
. 1 4 8 12 16 20
(End)
MATHEMATICA
f[0] = 1; f[n_] := 4 n; Array[f, 59, 0] (* or *)
CoefficientList[ Series[(1 + x)^2/(1 - x)^2, {x, 0, 58}], x] (* Robert G. Wilson v, Jan 02 2011 *)
PROG
(Haskell)
a008574 0 = 1; a008574 n = 4 * n
CROSSREFS
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. Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Expansion of (x^4+3*x^3+6*x^2+3*x+1)/((1-x)*(1-x^3)).
+10
57
1, 4, 10, 14, 18, 24, 28, 32, 38, 42, 46, 52, 56, 60, 66, 70, 74, 80, 84, 88, 94, 98, 102, 108, 112, 116, 122, 126, 130, 136, 140, 144, 150, 154, 158, 164, 168, 172, 178, 182, 186, 192, 196, 200, 206, 210, 214, 220, 224, 228, 234, 238, 242, 248, 252, 256, 262
COMMENTS
Coordination sequence for Dual(3^3.4^2) tiling with respect to a tetravalent node. This tiling is also called the prismatic pentagonal tiling, or the cem-d net. It is one of the 11 Laves tilings. (The identification of this coordination sequence with the g.f. in the definition was first conjectured by Colin Barker, Jan 22 2018.) Also, coordination sequence for a tetravalent node in the "krl" 2-D tiling (or net).
Both of these identifications are easily established using the "coloring book" method - see the Goodman-Strauss & Sloane link.
For n>0, this is twice A047386 (numbers congruent to 0 or +-2 mod 7). Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, 3rd row, second tiling. (For the krl tiling.)
B. Gruenbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987. See p. 96. (For the Dual(3^3.4^2) tiling.)
LINKS
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).
FORMULA
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. (Conjectured, correctly, by Colin Barker, Jan 22 2018.)
MATHEMATICA
CoefficientList[Series[(x^4+3x^3+6x^2+3x+1)/((1-x)(1-x^3)), {x, 0, 60}], x] (* or *) LinearRecurrence[{1, 0, 1, -1}, {1, 4, 10, 14, 18}, 80] (* Harvey P. Dale, Oct 03 2018 *)
PROG
(PARI) See Links section.
CROSSREFS
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. Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Coordination sequence for 3.3.4.3.4 Archimedean tiling.
+10
54
1, 5, 11, 16, 21, 27, 32, 37, 43, 48, 53, 59, 64, 69, 75, 80, 85, 91, 96, 101, 107, 112, 117, 123, 128, 133, 139, 144, 149, 155, 160, 165, 171, 176, 181, 187, 192, 197, 203, 208, 213, 219, 224, 229, 235, 240, 245, 251, 256, 261, 267, 272, 277, 283, 288, 293, 299
COMMENTS
a(n) is the number of vertices of the 3.3.4.3.4 tiling (which has three triangles and two squares, in the given cyclic order, meeting at each vertex) whose shortest path connecting them to a given origin vertex contains n edges.
First few terms provided by Allan C. Wechsler; Fred Lunnon and Fred Helenius gave the next few; Fred Lunnon suggested that the recurrence was a(n+3) = a(n) + 16 for n > 1. [This conjecture is true - see the CGS-NJAS link for a proof. - N. J. A. Sloane, Dec 31 2017] Appears also to be coordination sequence for node of type V2 in "krd" 2-D tiling (or net). This should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 25 2018 Appears also to be coordination sequence for node of type V1 in "krj" 2-D tiling (or net). This also should be easy to prove by the coloring book method (see link). - N. J. A. Sloane, Mar 26 2018
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 1st row, 2nd tiling, also 2nd row, third tiling.
LINKS
Reticular Chemistry Structure Resource, ttsN. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
FORMULA
Conjectured to be a(n) = floor((16n+1)/3) for n>0; a(0) = 1; this is a consequence of the suggested recurrence due to Lunnon (see comments). [This conjecture is true - see the CGS-NJAS link in A296368 for a proof. - N. J. A. Sloane, Dec 31 2017] a(n) = (16*n - ChebyshevU(n-1, -1/2))/3 for n>0 with a(0)=1.
MAPLE
A219529:= n -> `if`(n=0, 1, (16*n +1 - `mod`(n+1, 3))/3);
MATHEMATICA
Join[{1}, LinearRecurrence[{1, 0, 1, -1}, {5, 11, 16, 21}, 60]] (* Jean-François Alcover, Dec 13 2018 *) Table[If[n==0, 1, (16*n +1 - Mod[n+1, 3])/3], {n, 0, 60}] (* G. C. Greubel, May 27 2020 *) CoefficientList[Series[(x+1)^4/((x^2+x+1)(x-1)^2), {x, 0, 70}], x] (* Harvey P. Dale, Jul 03 2021 *)
PROG
(Haskell)
-- Very slow, could certainly be accelerated. SST stands for Snub Square Tiling.
setUnion [] l2 = l2
setUnion (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
where doRest = setUnion rst l2
setDifference [] l2 = []
setDifference (a:rst) l2 = if (elem a l2) then doRest else (a:doRest)
where doRest = setDifference rst l2
adjust k = (if (even k) then 1 else -1)
weirdAdjacent (x, y) = (x+(adjust y), y+(adjust x))
sstAdjacents (x, y) = [(x+1, y), (x-1, y), (x, y+1), (x, y-1), (weirdAdjacent (x, y))]
sstNeighbors core = foldl setUnion core (map sstAdjacents core)
sstGlob n core = if (n == 0) then core else (sstGlob (n-1) (sstNeighbors core))
sstHalo core = setDifference (sstNeighbors core) core
origin = [(0, 0)]
a219529 n = length (sstHalo (sstGlob (n-1) origin))
(Sage) [1]+[(16*n+1 -(n+1)%3)/3 for n in (1..60)] # G. C. Greubel, May 27 2020
CROSSREFS
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
EXTENSIONS
Corrected attributions and epistemological status in Comments; provided slow Haskell code - Allan C. Wechsler, Nov 30 2012
a(0)=1, a(1)=4; thereafter a(n) = 4*n-2*(-1)^n.
+10
39
1, 4, 6, 14, 14, 22, 22, 30, 30, 38, 38, 46, 46, 54, 54, 62, 62, 70, 70, 78, 78, 86, 86, 94, 94, 102, 102, 110, 110, 118, 118, 126, 126, 134, 134, 142, 142, 150, 150, 158, 158, 166, 166, 174, 174, 182, 182, 190, 190, 198, 198, 206, 206, 214, 214, 222, 222, 230, 230, 238, 238
COMMENTS
Coordination sequence for the bew tiling with respect to a point where two hexagons meet at only a single point. The coordination sequence for the other type of point can be shown to be A008574. Notes: There is one point on the positive x-axis at edge-distance n from the origin iff n is even; there is one point on the positive y-axis at edge-distance n from the origin iff n>1 is odd; and the number of points inside the first quadrant at distance n from 0 is n if n is odd, and n-1 if n is even.
Then a(n) = 2*(number on positive x-axis + number on positive y-axis) + 4*(number in interior of first quadrant).
LINKS
N. J. A. Sloane, Illustration of initial terms. Points in the first quadrant are marked with their edge-distance from the origin (the heavy black circle). (Ignore the black rectangles, which show some fundamental cells for this tiling.)
FORMULA
G.f.: (1 + 3*x + x^2 + 5*x^3 - 2*x^4) / ((1 - x)^2*(1 + x)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.
(End)
MATHEMATICA
{1, 4}~Join~Array[4 # - 2 (-1)^# &, 59, 2] (* or *)
LinearRecurrence[{1, 1, -1}, {1, 4, 6, 14, 14}, 61] (* or *)
CoefficientList[Series[(1 + 3 x + x^2 + 5 x^3 - 2 x^4)/((1 - x)^2*(1 + x)), {x, 0, 60}], x] (* Michael De Vlieger, Dec 23 2017 *)
PROG
(PARI) Vec((1 + 3*x + x^2 + 5*x^3 - 2*x^4) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Dec 23 2017
CROSSREFS
Apart from first two terms, same as A168384. Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 3.4.6.4.
+10
38
1, 4, 6, 7, 10, 14, 20, 24, 24, 23, 26, 34, 42, 44, 40, 37, 42, 54, 64, 64, 56, 51, 58, 74, 86, 84, 72, 65, 74, 94, 108, 104, 88, 79, 90, 114, 130, 124, 104, 93, 106, 134, 152, 144, 120, 107, 122, 154, 174, 164, 136, 121, 138, 174, 196, 184, 152, 135, 154, 194, 218
COMMENTS
Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See page 67, 4th row, 3rd tiling.
Otto Krötenheerdt, Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene, I, II, III, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math-Natur. Reihe, 18 (1969), 273-290; 19 (1970), 19-38 and 97-122. [Includes classification of 2-uniform tilings]
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166.
FORMULA
Based on the b-file, the g.f. appears to be (-2*x^9+6*x^8-8*x^7+7*x^6-2*x^5-2*x^4+5*x^3-2*x^2+1) / (x^6-4*x^5+8*x^4-10*x^3+8*x^2-4*x+1). - N. J. A. Sloane, Dec 14 2015
MATHEMATICA
LinearRecurrence[{4, -8, 10, -8, 4, -1}, {1, 4, 6, 7, 10, 14, 20, 24, 24, 23}, 100] (* Paolo Xausa, Nov 15 2023 *)
CROSSREFS
See A265035 for the other type of point. Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Coordination sequence for node of type 3.12.12 in "cph" 2-D tiling (or net).
+10
38
1, 3, 6, 7, 8, 15, 18, 17, 20, 25, 28, 29, 30, 35, 40, 39, 40, 47, 50, 49, 52, 57, 60, 61, 62, 67, 72, 71, 72, 79, 82, 81, 84, 89, 92, 93, 94, 99, 104, 103, 104, 111, 114, 113, 116, 121, 124, 125, 126, 131, 136, 135, 136, 143, 146, 145, 148, 153, 156, 157, 158
COMMENTS
Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, bottom row, first tiling.
FORMULA
G.f. = -(2*x^8-2*x^7-x^6-4*x^5-2*x^4-2*x^3-4*x^2-2*x-1) / ((x^2+1)*(x^2+x+1)*(x-1)^2). N. J. A. Sloane, Mar 28 2018 (This is now a theorem. - N. J. A. Sloane, Apr 05 2018) Theorem: G.f. = (1+2*x+4*x^2+2*x^3+2*x^4+4*x^5+1*x^6+2*x^7-2*x^8) / ((1-x)*(1+x^2)*(1-x^3)).
Proof. This follows by applying the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the links, and the details of the proof (by calculating the generating function) are on the next two scanned pages. - N. J. A. Sloane, Apr 05 2018
MATHEMATICA
Join[{1, 3, 6}, LinearRecurrence[{1, -1, 2, -1, 1, -1}, {7, 8, 15, 18, 17, 20}, 100]] (* Jean-François Alcover, Aug 05 2018 *)
PROG
(PARI) See Links section.
CROSSREFS
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Coordination sequence for a tetravalent node of type 3.4.3.12 in "cph" 2-D tiling (or net).
+10
38
1, 4, 5, 6, 12, 14, 15, 18, 21, 26, 28, 26, 31, 38, 37, 38, 44, 46, 47, 50, 53, 58, 60, 58, 63, 70, 69, 70, 76, 78, 79, 82, 85, 90, 92, 90, 95, 102, 101, 102, 108, 110, 111, 114, 117, 122, 124, 122, 127, 134, 133, 134, 140, 142, 143, 146, 149, 154, 156, 154
COMMENTS
Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 31 2023
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 66, bottom row, first tiling.
FORMULA
Theorem: G.f. = (1+2*x+4*x^3+3*x^4+6*x^6-4*x^7+6*x^8-2*x^9) / ((1-x)^2*(1+x^2)*(1+x^2+x^4)).
The proof uses the coloring book method described in the Goodman-Strauss & Sloane article. The trunks and branches structure is shown in the first scan. (Not yet added.) The trunks are blue, the branches are red, and the twigs are green. There is mirror symmetry about the Y-axis, and quadrants I and II are essentially identical, as are quadrants III and IV. The counts of the various classes of nodes are given in the second scan, and the corresponding generating functions are in the third scan. Adding up the different terms gives the g.f. stated above. - N. J. A. Sloane, Apr 07 2018 E.g.f.: (6*(3 + (4*exp(x) - 3)*x + 3*sin(x)) - 9*cos(sqrt(3)*x/2)*cosh(x/2) + sqrt(3)*sin(sqrt(3)*x/2)*(8*cosh(x/2) - 5*sinh(x/2)))/9. - Stefano Spezia, Jun 08 2024
MATHEMATICA
Join[{1, 4}, LinearRecurrence[{2, -3, 4, -4, 4, -3, 2, -1}, {5, 6, 12, 14, 15, 18, 21, 26}, 100]] (* Jean-François Alcover, Aug 05 2018 *)
PROG
(PARI) \\ See Links section.
CROSSREFS
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Expansion of (x^4+3*x^3+x^2+3*x+1)/((x^2+1)*(x-1)^2).
+10
38
1, 5, 9, 13, 18, 23, 27, 31, 36, 41, 45, 49, 54, 59, 63, 67, 72, 77, 81, 85, 90, 95, 99, 103, 108, 113, 117, 121, 126, 131, 135, 139, 144, 149, 153, 157, 162, 167, 171, 175, 180, 185, 189, 193, 198, 203, 207, 211, 216, 221, 225, 229, 234, 239, 243, 247, 252
COMMENTS
Appears to be coordination sequence for node of type 3^3.4^2 in "krm" 2-D tiling (or net).
Also appears to be coordination sequence for pentavalent node in "krk" 2-D tiling (or net).
Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, row 3, first tiling; also p. 66, row 3, first tiling.
FORMULA
For explicit formula for a(n) see Maple code.
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n > 4. - Colin Barker, Mar 23 2018
MAPLE
f:=proc(n) if n=0 then 1
elif (n mod 2) = 0 then 9*n/2
elif (n mod 4) = 1 then 18*(n-1)/4+5
else 18*(n-3)/4+13; fi; end;
s1:=[seq(f(n), n=0..60)];
MATHEMATICA
Join[{1}, LinearRecurrence[{2, -2, 2, -1}, {5, 9, 13, 18}, 60]] (* Jean-François Alcover, Jan 08 2019 *)
PROG
(PARI) Vec((x^4+3*x^3+x^2+3*x+1)/((x^2+1)*(x-1)^2) + O(x^60)) \\ Colin Barker, Mar 23 2018
CROSSREFS
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
Expansion of (x^2+x+1)^2 / ((x^2+1)*(x-1)^2).
+10
38
1, 4, 9, 14, 18, 22, 27, 32, 36, 40, 45, 50, 54, 58, 63, 68, 72, 76, 81, 86, 90, 94, 99, 104, 108, 112, 117, 122, 126, 130, 135, 140, 144, 148, 153, 158, 162, 166, 171, 176, 180, 184, 189, 194, 198, 202, 207, 212, 216, 220, 225, 230, 234, 238, 243, 248, 252
COMMENTS
Appears to be coordination sequence for node of type 4^4 in "krm" 2-D tiling (or net).
Also appears to be coordination sequence for tetravalent node in "krk" 2-D tiling (or net).
Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023
REFERENCES
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, row 3, first tiling; also p. 66, row 3, first tiling.
FORMULA
For explicit formula for a(n) see Maple code.
a(n) = 9*n/2 + (1 - (-1)^n)*i^(n*(n + 1))/4 for n>0, a(0)=1 and i=sqrt(-1). Therefore, for even n>0 a(n) = 9*n/2, otherwise a(n) = 9*n/2 - (-1)^((n-1)/2)/2. - Bruno Berselli, Mar 23 2018 a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4. - Colin Barker, Mar 23 2018
MAPLE
f:=proc(n) if n=0 then 1
elif (n mod 2) = 0 then 9*n/2
elif (n mod 4) = 1 then 18*(n-1)/4+4
else 18*(n-3)/4+14; fi; end;
s1:=[seq(f(n), n=0..60)];
MATHEMATICA
Join[{1}, LinearRecurrence[{2, -2, 2, -1}, {4, 9, 14, 18}, 60]] (* Jean-François Alcover, Jan 08 2019 *)
PROG
(PARI) Vec((x^2+x+1)^2 / ((x^2+1)*(x-1)^2) + O(x^60)) \\ Colin Barker, Mar 23 2018
CROSSREFS
Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.
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