A008574 -id:A008574 - cp's OEIS frontend

A329501 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n squares cut from the square grid by cuts parallel to grid lines.

1, 1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 4, 6, 4, 2, 1, 4, 7, 6, 4, 2, 1, 4, 8, 8, 6, 4, 2, 1, 4, 8, 10, 8, 6, 4, 2, 1, 4, 8, 11, 10, 8, 6, 4, 2, 1, 4, 8, 12, 12, 10, 8, 6, 4, 2, 1, 4, 8, 12, 14, 12, 10, 8, 6, 4, 2

Offset: 1

N. J. A. Sloane, Nov 19 2019

For the case when the cuts are at 45 degrees to the grid lines, see A329504.
See A329508, A329512, and A329515 for coordination sequences for cylinders formed by rolling up the hexagonal grid ("carbon nanotubes").
The g.f.s for the rows can easily be found using the "trunks and branches" method (see Goodman-Strauss and Sloane). In the illustration for n=5, there are two trunks (blue) and ten branches (red).

			Array begins:
  1, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
  1, 3, 4,  4,  4,  4,  4,  4,  4,  4,  4,  4, ...
  1, 4, 6,  6,  6,  6,  6,  6,  6,  6,  6,  6, ...
  1, 4, 7,  8,  8,  8,  8,  8,  8,  8,  8,  8, ...
  1, 4, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...
  1, 4, 8, 11, 12, 12, 12, 12, 12, 12, 12, 12, ...
  1, 4, 8, 12, 14, 14, 14, 14, 14, 14, 14, 14, ...
  1, 4, 8, 12, 15, 16, 16, 16, 16, 16, 16, 16, ...
  1, 4, 8, 12, 16, 18, 18, 18, 18, 18, 18, 18, ...
  1, 4, 8, 12, 16, 19, 20, 20, 20, 20, 20, 20, ...
  ...
The initial antidiagonals are:
  1;
  1,  2;
  1,  3,  2;
  1,  4,  4,  2;
  1,  4,  6,  4,  2;
  1,  4,  7,  6,  4,  2;
  1,  4,  8,  8,  6,  4,  2;
  1,  4,  8, 10,  8,  6,  4,  2;
  1,  4,  8, 11, 10,  8,  6,  4,  2;
  1,  4,  8, 12, 12, 10,  8,  6,  4,  2;
  1,  4,  8, 12, 14, 12, 10,  8,  6,  4,  2;
  ...

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, Illustration for rows 1 through 5, showing vertices of cylinder labeled with distance from base point (c = n is the width (or circumference)). The cylinders are formed by identifying the black lines.
Index entries for coordination sequences

Rows 1,2,3,4,5 are A040000, A113311, A329502, A115291, A329503.

Cf. A008574, A329504-A329517.

Let theta = (1+x)/(1-x).
If n = 2*k, the g.f. for the coordination sequence for row n is theta*(1+2*x+2*x^2+...+2*x^(k-1)+x^k).
If n = 2*k+1, the g.f. for the coordination sequence for row n is theta*(1+2*x+2*x^2+...+2*x^k).

A054275 Susceptibility series H_2 for 2-dimensional Ising model (divided by 2).

Original entry on oeis.org

1, 8, 24, 52, 90, 140, 200, 272, 354, 448, 552, 668, 794, 932, 1080, 1240, 1410, 1592, 1784, 1988, 2202, 2428, 2664, 2912, 3170, 3440, 3720, 4012, 4314, 4628, 4952, 5288, 5634, 5992, 6360, 6740, 7130, 7532, 7944, 8368, 8802, 9248, 9704, 10172, 10650, 11140

Offset: 0

N. J. A. Sloane, May 09 2000

Colin Barker, Table of n, a(n) for n = 0..1000
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A008574, A054410, A054389, A054764.

Concatenation([1], List([1..50], n-> (22*n^2+9-(-1)^n)/4)); # G. C. Greubel, Jul 31 2019

[n eq 0 select 1 else (22*n^2+9-(-1)^n)/4: n in [0..50]]; // G. C. Greubel, Jul 31 2019

CoefficientList[Series[(1+6*x+8*x^2+6*x^3+x^4)/((1-x)^3*(1+x)),{x,0,50}], x] (* or *) LinearRecurrence[{2,0,-2,1},{1,8,24,52,90},51] (* Indranil Ghosh, Feb 24 2017 *)
Table[If[n==0, 1, (22*n^2+9-(-1)^n)/4], {n,0,50}] (* G. C. Greubel, Jul 31 2019 *)

Vec((1+6*x+8*x^2+6*x^3+x^4)/((1-x)^3*(1+x)) + O(x^50)) \\ Colin Barker, Dec 09 2016

a(n)=if(n, 11*n^2+5, 2)\2 \\ Charles R Greathouse IV, Feb 24 2017

[1]+[(22*n^2+9-(-1)^n)/4 for n in (1..50)] # G. C. Greubel, Jul 31 2019

G.f.: (1+6*x+8*x^2+6*x^3+x^4) / ((1-x)^3*(1+x)).
From Colin Barker, Dec 09 2016: (Start)
a(n) = (11*n^2+4)/2 for n>0 and even.
a(n) = (11*n^2+5)/2 for n odd.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n>4. (End)
E.g.f.: ((9 + 22*x + 22*x^2)*exp(x) - 4 - exp(-x))/4. - G. C. Greubel, Jul 31 2019

A054389 Susceptibility series H_5 for 2-dimensional Ising model (divided by 2).

Original entry on oeis.org

1, 20, 140, 620, 2016, 5364, 12292, 25228, 47488, 83508, 138908, 220748, 337568, 499668, 719124, 1010092, 1388800, 1873876, 2486316, 3249836, 4190816, 5338676, 6725796, 8387916, 10364032, 12696820, 15432508, 18621324, 22317344, 26578964, 31468724, 37053804

Offset: 0

N. J. A. Sloane, May 09 2000

Colin Barker, Table of n, a(n) for n = 0..1000
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,10,-4,-4,4,-1).

Cf. A008574, A054275, A054410, A054764.

Concatenation([1], List([1..35], n-> n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120)); # G. C. Greubel, Jul 31 2019

[1] cat [n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120: n in [1..35]]; // G. C. Greubel, Jul 31 2019

LinearRecurrence[{4,-4,-4,10,-4,-4,4,-1}, {1,20,140,620,2016,5364,12292, 25228,47488},35] (* or *) CoefficientList[Series[(1 +16*x +64*x^2 + 144*x^3 +166*x^4 +144*x^5 +64*x^6 +16*x^7 +x^8)/((1-x)^6*(1+x)^2), {x,0, 35}], x] (* Indranil Ghosh, Feb 24 2017 *)
Table[If[n==0, 1, n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120], {n,0,35}] (* G. C. Greubel, Jul 31 2019 *)

Vec((1 +16*x +64*x^2 +144*x^3 +166*x^4 +144*x^5 +64*x^6 +16*x^7 + x^8)/((1-x)^6*(1+x)^2) + O(x^35)) \\ Colin Barker, Dec 09 2016

[1]+[n*(154*n^4 +1260*n^2 +941 -45*(-1)^n)/120 for n in (1..35)] # G. C. Greubel, Jul 31 2019

G.f.: (1 + 16*x + 64*x^2 + 144*x^3 + 166*x^4 + 144*x^5 + 64*x^6 + 16*x^7 + x^8) / ((1 - x)^6*(1 + x)^2).
From Colin Barker, Dec 09 2016: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) - 4*a(n-3) + 10*a(n-4) - 4*a(n-5) - 4*a(n-6) + 4*a(n-7) - a(n-8) for n>8.
a(n) = (77*n^5 + 630*n^3 + 448*n)/60 for n>0 and even.
a(n) = (77*n^5 + 630*n^3 + 493*n)/60 for n odd. (End)
From G. C. Greubel, Jul 31 2019: (Start)
a(n) = n*(154*n^4 + 1260*n^2 + 941 - 45*(-1)^n)/120, n>0, with a(0)=1.
E.g.f.: (x*(2355 + 6090*x + 5110*x^2 + 1540*x^3 + 154*x^4)*exp(x) + 120 + 45*x*exp(-x))/120. (End)

A054410 Susceptibility series H_3 for 2-dimensional Ising model (divided by 2).

Original entry on oeis.org

1, 12, 52, 148, 328, 620, 1052, 1652, 2448, 3468, 4740, 6292, 8152, 10348, 12908, 15860, 19232, 23052, 27348, 32148, 37480, 43372, 49852, 56948, 64688, 73100, 82212, 92052, 102648, 114028, 126220, 139252, 153152, 167948, 183668, 200340, 217992, 236652

Offset: 0

N. J. A. Sloane, May 09 2000

Colin Barker, Table of n, a(n) for n = 0..1000
A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
D. Hansel et al., Analytical properties of the anisotropic cubic Ising model, J. Stat. Phys., 48 (1987), 69-80.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A008574, A054275, A054389, A054764.

Concatenation([1], List([1..40], n-> 2*n*(11+7*n^2)/3)); # G. C. Greubel, Jul 31 2019

[1] cat [2*n*(11+7*n^2)/3: n in [1..40]]; // G. C. Greubel, Jul 31 2019

CoefficientList[Series[(1+8*x+10*x^2+8*x^3+x^4)/(1-x)^4, {x,0,40}],x] (* or *) a[0]=1; a[n_]:= 2*n*(11+7*n^2)/3; Table[a[n], {n,0,40}] (* Indranil Ghosh, Feb 24 2017 *)
LinearRecurrence[{4,-6,4,-1},{1,12,52,148,328},50] (* Harvey P. Dale, Nov 24 2024 *)

Vec((1+8*x+10*x^2+8*x^3+x^4)/(1-x)^4 + O(x^40)) \\ Colin Barker, Dec 09 2016

vector(40, n, n--; if(n==0,1, 2*n*(11+7*n^2)/3)) \\ G. C. Greubel, Jul 31 2019

def A054410(n):
    if n == 0: return 1
    return 2*(n*(11 + 7*n**2))/3 # Indranil Ghosh, Feb 24 2017

[1]+[2*n*(11+7*n^2)/3 for n in (1..40)] # G. C. Greubel, Jul 31 2019

G.f.: (1 +8*x +10*x^2 +8*x^3 +x^4)/(1-x)^4.
From Colin Barker, Dec 09 2016: (Start)
a(n) = 2*n*(11 + 7*n^2)/3 for n>0.
a(0)=1, a(1)=12, a(2)=52, a(3)=148, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4. (End)
E.g.f.: (3 + 2*x*(18 + 21*x + 7*x^2)*exp(x))/3. - G. C. Greubel, Jul 31 2019

A008416 Coordination sequence for 8-dimensional cubic lattice.

Original entry on oeis.org

1, 16, 128, 688, 2816, 9424, 27008, 68464, 157184, 332688, 658048, 1229360, 2187520, 3732560, 6140800, 9785072, 15158272, 22900496, 33830016, 48978352, 69629696, 97364944, 134110592, 182192752, 244396544, 324031120, 425000576

Offset: 0

N. J. A. Sloane

Coordination sequence for 8-dimensional cyclotomic lattice Z[zeta_16].

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv:math/0508136 [math.CO], 2005-2006.
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
Ross McPhedran, Numerical Investigations of the Keiper-Li Criterion for the Riemann Hypothesis, arXiv:2311.06294 [math.NT], 2023. See p. 6.
Index entries for linear recurrences with constant coefficients, signature (8, -28, 56, -70, 56, -28, 8, -1).

Cf. A008574, A005899, A008412, A008413, A008414, A008415, A008418, A008420.

Cf. A113413, A122542.

CoefficientList[Series[((1 + x)/(1 - x))^8, {x, 0, 26}], x] (* Michael De Vlieger, Dec 18 2017 *)

G.f.: ((1+x)/(1-x))^8.
a(n) = A008415(n) + A008415(n-1) + a(n-1). - Bruce J. Nicholson, Dec 17 2017
n*a(n) = 16*a(n-1) + (n-2)*a(n-2) for n > 1. - Seiichi Manyama, Jun 06 2018

A102301 a(n) = ((3n + 1)2^(n+3) + 9 + (-1)^n)/18.

Original entry on oeis.org

1, 4, 13, 36, 93, 228, 541, 1252, 2845, 6372, 14109, 30948, 67357, 145636, 313117, 669924, 1427229, 3029220, 6407965, 13514980, 28428061, 59652324, 124897053, 260978916, 544327453, 1133394148, 2356266781, 4891490532, 10140895005, 20997617892, 43426891549

Offset: 0

Creighton Dement, Feb 20 2005

A floretion-generated sequence resulting from particular transform of A000975.

Floretion Algebra Multiplication Program, FAMP Code: 2jesforseq[ + .5'i + 'kk' + .5'jk' ], 1vesforseq(n) = A000975(n+2)*(-1)^(n+1), ForType: 1A, LoopType: tes (2nd iteration)

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. Etzion, On the stopping redundancy of Reed-Muller codes, IEEE Trans. Information Theory 52 (11) (2006) 4867-4879, also, arXiv:cs/0511056 [cs.IT], 2005.
Toufik Mansour and Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).

Cf. A000975, A000337, A045883, A001787, A059570, A137266, A008574, A059260.

[((3*n+1)*2^(n+3)+9+(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Nov 21 2018

Table[((3n+1)*2^(n+3) + 9 + (-1)^n)/18, {n,0,50}] (* G. C. Greubel, Sep 27 2017 *)
LinearRecurrence[{4, -3, -4, 4}, {1, 4, 13, 36}, 50] (* Vincenzo Librandi, Nov 21 2018 *)

a(n)=((3*n+1)*2^(n+3)+9+(-1)^n)/18 \\ Charles R Greathouse IV, Oct 16 2015

G.f.: 1/((1-x^2)*(1-2*x)^2).
a(n+1) - 2*a(n) = A000975(n+2) (n-th number without consecutive equal binary digits)
a(n) + a(n+1) = A000337(n+2);
a(n+1) - a(n) = A045883(n+2);
a(n+2) - a(n) = A001787(n+3) ( Number of edges in n-dimensional hypercube );
a(n+2) - 2*a(n+1) + a(n) = A059570(n+3);
Convolution of "Number of fixed points in all 231-avoiding involutions in S_n" (A059570) with the natural numbers (A000027), treating the result as if offset=0. - Graeme McRae, Jul 12 2006
Equals triangle A059260 * A008574 as a vector, where A008574 = [1, 4, 8, 12, 16, 20, ...]. - Gary W. Adamson, Mar 06 2012

A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 5, 14, 32, 67, 133, 256, 484, 905, 1681, 3110, 5740, 10579, 19481, 35856, 65976, 121377, 223277, 410702, 755432, 1389491, 2555709, 4700720, 8646012, 15902537, 29249369, 53798022, 98950036, 181997539, 334745713

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+2) represent the Kn13 and Kn23 sums of the square array of Delannoy numbers A008288. See A180662 for the definition of these knight and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (3,-2,0,-1,1).

Cf. A000073 (Kn11 & Kn21), A089068 (Kn12 & Kn22), A180668 (Kn13 & Kn23), A180669 (Kn14 & Kn24), A180670 (Kn15 & Kn25).

nmax:=31: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):= a(n-1)+a(n-2)+a(n-3)+4*n-8 od: seq(a(n),n=0..nmax);

LinearRecurrence[{3,-2,0,-1,1},{0,0,1,5,14},40] (* Harvey P. Dale, Dec 15 2023 *)

a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.
a(n) = a(n-1)+A001590(n+3)-2 with a(0)=0.
a(n) = sum(A008574(m)*A000073(n-m),m=0..n).
a(n+2) = add(A008288(n-k+2,k+2),k=0..floor(n/2)).
GF(x) = (x^2*(1+x)^2)/((1-x)^2*(1-x-x^2-x^3)).
Contribution from Bruno Berselli, Sep 23 2010: (Start)
a(n) = 2*a(n-1)-a(n-4)+4 for n>4.
a(n)-3*a(n-1)+2a(n-2)+a(n-4)-a(n-5) = 0 for n>4. (End)

A319840 Table read by antidiagonals: T(n, k) is the number of elements on the perimeter of an n X k matrix.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 8, 8, 5, 6, 10, 10, 10, 10, 6, 7, 12, 12, 12, 12, 12, 7, 8, 14, 14, 14, 14, 14, 14, 8, 9, 16, 16, 16, 16, 16, 16, 16, 9, 10, 18, 18, 18, 18, 18, 18, 18, 18, 10, 11, 20, 20, 20, 20, 20, 20, 20, 20, 20, 11, 12, 22, 22, 22

Offset: 1

Stefano Spezia, Sep 29 2018

The table T(n, k) can be indifferently read by ascending or descending antidiagonals.

			The table T starts in row n=1 with columns k >= 1 as:
   1   2   3   4   5   6   7   8   9  10 ...
   2   4   6   8  10  12  14  16  18  20 ...
   3   6   8  10  12  14  16  18  20  22 ...
   4   8  10  12  14  16  18  20  22  24 ...
   5  10  12  14  16  18  20  22  24  26 ...
   6  12  14  16  18  20  22  24  26  28 ...
   7  14  16  18  20  22  24  26  28  30 ...
   8  16  18  20  22  24  26  28  30  32 ...
   9  18  20  22  24  26  28  30  32  34 ...
  10  20  22  24  26  28  30  32  34  36 ...
  ...
The triangle X(n, k) begins
  n\k|   1   2   3   4   5   6   7   8   9  10
  ---+----------------------------------------
   1 |   1
   2 |   2   2
   3 |   3   4   3
   4 |   4   6   6   4
   5 |   5   8   8   8   5
   6 |   6  10  10  10  10   6
   7 |   7  12  12  12  12  12   7
   8 |   8  14  14  14  14  14  14   8
   9 |   9  16  16  16  16  16  16  16   9
  10 |  10  18  18  18  18  18  18  18  18  10
  ...

Cf. A000027 (1st column/right diagonal of the triangle or 1st row/column of the table), A005843 (2nd row/column of the table, or 2nd column of the triangle), A008574 (main diagonal of the table), A005893 (row sum of the triangle).
Cf. A003991 (the number of elements in an n X k matrix).
Cf. A131821, A318274, A154325.

[[k lt 3 or n+1-k lt 3 select (n+1-k)*k else 2*n-2: k in [1..n]]: n in [1..10]]; // triangle output

a := (n, k) -> (n+1-k)*k-(n-1-k)*(k-2)*(limit(Heaviside(min(n+1-k, k)-3+x), x = 0, right)): seq(seq(a(n, k), k = 1 .. n), n = 1 .. 20)

Flatten[Table[(n + 1 - k) k-(n-1-k)*(k-2)Limit[HeavisideTheta[Min[n+1-k,k]-3+x], x->0, Direction->"FromAbove"  ],{n, 20}, {k, n}]] (* or *)
f[n_] := Table[SeriesCoefficient[(x y - x^3 y^3)/((-1 + x)^2 (-1 + y)^2), {x, 0, i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f,20]]

T(n, k) = if ((n+1-k<3) || (k<3), (n+1-k)*k, 2*n-2);
tabl(nn) = for(i=1, nn, for(j=1, i, print1(T(i, j), ", ")); print);
tabl(20) \\ triangle output

T(n, k) = n*k - (n - 2)*(k - 2)*H(min(n, k) - 3), where H(x) is the Heaviside step function, taking H(0) = 1.
G.f. as rectangular array: (x*y - x^3*y^3)/((-1 + x)^2*(-1 + y)^2).
X(n, k) = A131821(n, k)*A318274(n - 1, k)*A154325(n - 1, k). - Franck Maminirina Ramaharo, Nov 18 2018

A336627 Coordination sequence for the Manhattan lattice.

Original entry on oeis.org

1, 2, 4, 8, 11, 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

Offset: 0

Sean A. Irvine, Jul 28 2020

In the Manhattan lattice, N-S streets run alternately N and S, and E-W streets run alternately E and W. - N. J. A. Sloane, Jul 29 2020

Sean A. Irvine, Illustration of a(0) to a(7)
N. J. A. Sloane, Crude drawing of initial layers showing paths of length 6 from origin (looking North-West). The presence of three points at distance 4 from the origin on the line of symmetry explains why a(4) is odd!
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008574 (square lattice), A117633 (self-avoiding walks).

CoefficientList[Series[(1+x^2)(1+2x^3-x^4)/(1-x)^2,{x,0,80}],x] (* or *) LinearRecurrence[{2,-1},{1,2,4,8,11,16,20},80] (* Harvey P. Dale, Dec 28 2021 *)

a(n)=if(n>4, 4*n-4, min(2^n, 11)) \\ Charles R Greathouse IV, Oct 18 2022

G.f.: (1+x^2) * (1+2*x^3-x^4) / (1-x)^2.

a(n) = 4*(n-1), n >= 5.

A068600 Number of n-uniform tilings having n different arrangements of polygons about their vertices.

Original entry on oeis.org

11, 20, 39, 33, 15, 10, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Brian Galebach, Mar 28 2002

nonn

Sequence gives the number of edge-to-edge regular-polygon tilings having n topologically distinct vertex types, with each vertex type having a different arrangement of surrounding polygons. Does not allow for tilings with two or more vertex types having the same arrangement of surrounding polygons, even when those vertices are topologically distinct. There are no 8- or higher-uniform tilings having the equivalent number of distinct polygon arrangements.

There are eleven 1-uniform tilings (also called the "Archimedean" tessellations) which comprise the three regular tessellations (all triangles, squares, or hexagons) plus the eight semiregular tessellations. (See A250120. - N. J. A. Sloane, Nov 29 2014)

This sequence was originally calculated by Otto Krotenheerdt.
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, page 69.
Krotenheerdt, Otto. "Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene," Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-natur. Reihe, 18(1969), 273-290; 19 (1970)19-38 and 97-122.

Steven Dutch, Uniform Tilings
Brian L. Galebach, n-Uniform Tilings
Ng Lay Ling, Honours Project - Tilings and Patterns.

Cf. A068599.

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).

cp's OEIS Frontend

A329501 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n squares cut from the square grid by cuts parallel to grid lines.

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A054275 Susceptibility series H_2 for 2-dimensional Ising model (divided by 2).

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A054389 Susceptibility series H_5 for 2-dimensional Ising model (divided by 2).

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A054410 Susceptibility series H_3 for 2-dimensional Ising model (divided by 2).

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A008416 Coordination sequence for 8-dimensional cubic lattice.

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A102301 a(n) = ((3*n + 1)*2^(n+3) + 9 + (-1)^n)/18.

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A180668 a(n) = a(n-1)+a(n-2)+a(n-3)+4*n-8 with a(0)=0, a(1)=0 and a(2)=1.

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A102301 a(n) = ((3n + 1)2^(n+3) + 9 + (-1)^n)/18.