Displaying 1-10 of 56 results found.
1, 5, 15, 29, 47, 71, 99, 131, 169, 211, 257, 309, 365, 425, 491, 561, 635, 715, 799, 887, 981, 1079, 1181, 1289, 1401, 1517, 1639, 1765, 1895, 2031, 2171, 2315, 2465, 2619, 2777, 2941, 3109, 3281, 3459, 3641, 3827, 4019, 4215, 4415, 4621, 4831, 5045, 5265
COMMENTS
Linear recurrence and g.f. confirmed by Shutov/Maleev link in A298024. - Ray Chandler, Aug 31 2023
FORMULA
G.f.: (1 + 3*x + 6*x^2 + 3*x^3 + x^4) / ((1 - x)^3*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n>4.
(End)
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 (1 + x + x^2)/(1 - x)^2.
+10
86
1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186
COMMENTS
Also the Engel expansion of exp^(1/3); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002 Coordination sequence for planar net 6^3 (the graphite net, or the graphene crystal) - that is, the number of atoms at graph distance n from any fixed atom. Also for the hcb or honeycomb net. - N. J. A. Sloane, Jan 06 2013, Mar 31 2018 Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_3].
Conjecture: This is also the maximum number of edges possible in a planar simple graph with n+2 vertices. - Dmitry Kamenetsky, Jun 29 2008 The conjecture is correct. Proof: For n=0 the theorem holds, the maximum planar graph has n+2=2 vertices and 1 edge. Now suppose that we have a connected planar graph with at least 3 vertices. If it contains a face that is not a triangle, we can add an edge that divides this face into two without breaking its planarity. Hence all maximum planar graphs are triangulations. Euler's formula for planar graphs states that in any planar simple graph with V vertices, E edges and F faces we have V+F-E=2. If all faces are triangles, then F=2E/3, which gives us E=3V-6. Hence for n>0 each maximum planar simple graph with n+2 vertices has 3n edges. - Michal Forisek, Apr 23 2009 a(n) = sum of natural numbers m such that n - 1 <= m <= n + 1. Generalization: If a(n,k) = sum of natural numbers m such that n - k <= m <= n + k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). - Jaroslav Krizek, Nov 18 2009 Integers n dividing a(n) = a(n-1) - a(n-2) with initial conditions a(0)=0, a(1)=1 (see A128834 with offset 0). - Thomas M. Bridge, Nov 03 2013 a(n) is conjectured to be the number of polygons added after n iterations of the polygon expansions (type A, B, C, D & E) shown in the Ngaokrajang link. The patterns are supposed to become the planar Archimedean net 3.3.3.3.3.3, 3.6.3.6, 3.12.12, 3.3.3.3.6 and 4.6.12 respectively when n - > infinity. - Kival Ngaokrajang, Dec 28 2014 Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I. - Ray Chandler, Nov 21 2016 Conjecture: let m = n + 2, p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p (see the Wikipedia link below), and f(m) = a(n) - q. Then f(m) would be the solution of the Thompson problem for all m in 3-space. - Sergey Pavlov, Feb 03 2017 Also, sequence defined by a(0)=1, a(1)=3, c(0)=2, c(1)=4; and thereafter a(n) = c(n-1) + c(n-2), and c consists of the numbers missing from a (see A001651). - Ivan Neretin, Mar 28 2017
REFERENCES
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.
LINKS
Reticular Chemistry Structure Resource, hcb
FORMULA
a(0) = 1; a(n) = 3*n = A008585(n), n >= 1. Euler transform of length 3 sequence [3, 0, -1]. - Michael Somos, Aug 04 2009
EXAMPLE
G.f. = 1 + 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + 24*x^8 + ...
Illustration of initial terms as triangles:
. 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 3 6 9 12 15
(End)
MATHEMATICA
CoefficientList[Series[(1 + x + x^2) / (1 - x)^2, {x, 0, 80}], x] (* Vincenzo Librandi, Nov 23 2014 *)
PROG
(PARI) {a(n) = if( n==0, 1, 3 * n)}; /* Michael Somos, May 05 2015 */ (Haskell)
a008486 0 = 1; a008486 n = 3 * 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 sequence for hexagonal lattice.
+10
59
1, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348
COMMENTS
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. It is also the planar net 3.3.3.3.3.3.
Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_6].
Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 20 ).
Also the Engel expansion of exp^(1/6); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002 Numbers k such that k+floor(k/2) | k*floor(k/2). - Wesley Ivan Hurt, Dec 01 2020
LINKS
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 ( pdf). Reticular Chemistry Structure Resource, hxl
FORMULA
G.f.: (1 + 4*x + x^2)/(1 - x)^2.
Equals binomial transform of [1, 5, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Jul 08 2008 G.f.: Hypergeometric2F1([3,-2], [1], -x/(1-x)). - Paul Barry, Sep 18 2008 n*a(1) + (n-1)*a(2) + (n-2)*a(3) + ... + 2*a(n-1) + a(n) = n^3. - Warren Breslow, Oct 28 2013
EXAMPLE
Illustration of initial terms:
. 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 o o o o o o o
. 6 o o o o o o
. 12 o o o o o
. 18
. 24
(End)
G.f. = 1 + 6*x + 12*x^2 + 18*x^3 + 24*x^4 + 30*x^5 + 36*x^6 + 42*x^7 + 48*x^8 + 54*x^9 + ...
PROG
(PARI) {a(n) = 6*n + (!n)};
(Maxima) makelist(if n=0 then 1 else 6*n, n, 0, 65); /* Martin Ettl, Nov 12 2012 */ (SageMath) [6*n+int(n==0) for n in range(66)] # G. C. Greubel, May 25 2023
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 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
Coordination sequence for root lattice B_2.
+10
39
1, 8, 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
COMMENTS
Equivalently, the coordination sequence for a point of degree 8 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 4 is given by A234275. - N. J. A. Sloane, Dec 28 2015 Number of points of L_infinity norm n in the simple square lattice Z^2. - N. J. A. Sloane, Apr 15 2008 Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0( 24 ).
Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (11;0) and (01;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, j1<j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev, Nov 11 2004 These numbers correspond to the number of primes in the shells of a prime spiral. In a(2) there are 8 primes surrounding 2 in a prime spiral. - Enoch Haga, Apr 06 2000
REFERENCES
J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 191.
FORMULA
a(n) = [x^(2*n)] ((1 + x)/(1 - x))^2.
G.f. for coordination sequence of B_n lattice: Sum_{i=0..n} binomial(2*n+1, 2*i)*z^i - 2*n*z*(1+z)^(n-1)/(1-z)^n. [Bacher et al.]
a(n) = (2*n+1)^2 - (2*n-1)^2. Binomial transform of [1, 7, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Dec 27 2007
EXAMPLE
1 + 8*x + 16*x^2 + 24*x^3 + 32*x^4 + 40*x^5 + 48*x^6 + 56*x^7 + ...
CROSSREFS
Apart from initial term, the same as A008590. 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.
AUTHOR
Michael Baake (mbaake(AT)sunelc3.tphys.physik.uni-tuebingen.de)
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.
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