A008574 -id:A008574 - cp's OEIS frontend

A130713 a(0)=a(2)=1, a(1)=2, a(n)=0 for n > 2.

1, 2, 1, 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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Curtz and Tanya Khovanova, Jul 01 2007

Self-convolution of A019590. Up to a sign the convolutional inverse of the natural numbers sequence. - Tanya Khovanova, Jul 14 2007

Iterated partial sums give the chain A130713 -> A113311 -> A008574 -> A001844 -> A005900 -> A006325 -> A033455 -> A259181, up to index. The k-th term of the n-th partial sums is (n^2-7n+14 + 4k(k+n-4))(k+n-4)!/(k-1)!/(n-1)!, for k > 3-n. Iterating partial sums in reverse (n-th differences with n zeros prepended) gives row (n+3) of A182533, modulo signs and trailing zeros. - Travis Scott, Feb 19 2023

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.

A130713:=n->binomial(2*n, n^2); seq(A130713(n), n=0..100); # Wesley Ivan Hurt, Mar 08 2014

Table[Binomial[2 n, n^2], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 08 2014 *)

G.f.: 1 + 2*x + x^2.

a(n) = binomial(2n, n^2). - Wesley Ivan Hurt, Mar 08 2014

A241204 Expansion of (1 + 2x)^2/(1 - 2x)^2.

Original entry on oeis.org

1, 8, 32, 96, 256, 640, 1536, 3584, 8192, 18432, 40960, 90112, 196608, 425984, 917504, 1966080, 4194304, 8912896, 18874368, 39845888, 83886080, 176160768, 369098752, 771751936, 1610612736, 3355443200, 6979321856, 14495514624, 30064771072, 62277025792

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 17 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 12.
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Subsequence of A008574.

R:=PowerSeriesRing(Integers(), 41); Coefficients(R!((1+2*x)^2/(1-2*x)^2));

A241204:= n->`if`(n=0, 1, 2^(n+2)*n); seq(A241204(n), n=0..20); # Wesley Ivan Hurt, Apr 22 2014

Table[2^(n+2)*n + Boole[n==0], {n,0,40}] (* G. C. Greubel, Jun 07 2023 *)
LinearRecurrence[{4,-4},{1,8,32},30] (* Harvey P. Dale, Jun 23 2025 *)

Vec((2*x+1)^2/(2*x-1)^2 + O(x^100)) \\ Colin Barker, Apr 22 2014

def A241204(i):
    if i==0: return 1
    else: return 2^(2+i)*i;
[A241204(n) for n in (0..30)] # Bruno Berselli, Apr 23 2014

a(n) = 2^(2+n)*n for n>0. - Colin Barker, Apr 23 2014
a(n) = 4*a(n-1)-4*a(n-2) for n>2. - Colin Barker, Apr 23 2014
From Amiram Eldar, Jan 13 2021: (Start)
Sum_{n>=1} 1/a(n) = log(2)/4.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(3/2)/4. (End)
E.g.f.: 1 + 8*x*exp(x). - G. C. Greubel, Jun 07 2023

A250123 Coordination sequence of point of type 3.3.4.3.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Original entry on oeis.org

1, 5, 8, 8, 11, 17, 25, 27, 24, 30, 38, 46, 47, 44, 46, 50, 64, 68, 65, 66, 70, 80, 80, 83, 87, 91, 100, 100, 99, 99, 109, 121, 121, 119, 119, 125, 133, 139, 140, 140, 145, 153, 155, 152, 158, 166, 174, 175, 172, 174, 178, 192, 196, 193, 194, 198, 208, 208, 211

Offset: 0

N. J. A. Sloane, Nov 29 2014

nonn

This tiling appears as an example in Connelly et al. (2014), Fig. 6 (the heavy black lines in the figures here are an artifact from that figure).

For the definition of k-uniform tiling see Section 2.2 of Chapter 2 of Grünbaum and Shephard (1987).

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Joseph Myers, Table of n, a(n) for n = 0..1000
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, arXiv:1301.0664 [math.MG], 2013.
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, Discrete Comput. Geom. 52 (2014), no. 4, 754--779. MR3279548.
Brian Galebach, Tiling 132 (in list of 4-uniform tilings).
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
N. J. A. Sloane, A portion of the 3-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}. The four black dots labeled P,Q,R,S show the four types of point. The present sequence is for a point of type P.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250124, A250125, A250126.

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).

Empirical g.f.: -(x+1)*(x^15 +3*x^14 -4*x^11 -6*x^10 -7*x^9 -4*x^8 -7*x^7 -11*x^6 -9*x^5 -7*x^4 -4*x^3 -4*x^2 -4*x -1) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Dec 02 2014

Galebach link from Joseph Myers, Nov 30 2014

Extended by Joseph Myers, Dec 02 2014

A250124 Coordination sequence of point of type 3.3.12.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Original entry on oeis.org

1, 4, 7, 10, 15, 16, 21, 29, 28, 34, 33, 40, 48, 45, 53, 51, 59, 65, 64, 72, 68, 78, 83, 83, 89, 87, 97, 100, 102, 107, 106, 114, 119, 121, 124, 125, 132, 138, 138, 143, 144, 149, 157, 156, 162, 161, 168, 176, 173, 181, 179, 187, 193, 192, 200, 196, 206, 211, 211

Offset: 0

N. J. A. Sloane, Nov 29 2014

nonn

This tiling appears as an example in Connelly et al. (2014), Fig. 6 (the heavy black lines in the figures here are an artifact from that figure).

For the definition of k-uniform tiling see Section 2.2 of Chapter 2 of Grünbaum and Shephard (1987).

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Joseph Myers, Table of n, a(n) for n = 0..1000
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, arXiv:1301.0664 [math.MG], 2013.
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, Discrete Comput. Geom. 52 (2014), no. 4, 754--779. MR3279548.
Brian Galebach, Tiling 132 (in list of 4-uniform tilings).
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
N. J. A. Sloane, A portion of the 3-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}. The four black dots labeled P,Q,R,S show the four types of point. The present sequence is for a point of type R.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250123, A250125, A250126.

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).

Empirical g.f.: -(3*x^14 -4*x^12 -4*x^11 -7*x^10 -12*x^9 -14*x^8 -21*x^7 -17*x^6 -15*x^5 -15*x^4 -10*x^3 -7*x^2 -4*x -1) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Dec 02 2014

Galebach link from Joseph Myers, Nov 30 2014

Extended by Joseph Myers, Dec 02 2014

A250125 Coordination sequence of point of type 3.4.3.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Original entry on oeis.org

1, 4, 6, 11, 13, 15, 23, 23, 33, 30, 33, 42, 41, 54, 46, 54, 58, 58, 73, 64, 75, 74, 79, 89, 81, 94, 92, 100, 105, 102, 110, 109, 119, 123, 123, 126, 130, 135, 140, 142, 144, 151, 151, 161, 158, 161, 170, 169, 182, 174, 182, 186, 186, 201, 192, 203, 202, 207, 217

Offset: 0

N. J. A. Sloane, Nov 29 2014

nonn

This tiling appears as an example in Connelly et al. (2014), Fig. 6 (the heavy black lines in the figures here are an artifact from that figure).

For the definition of k-uniform tiling see Section 2.2 of Chapter 2 of Grünbaum and Shephard (1987).

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Joseph Myers, Table of n, a(n) for n = 0..1000
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, arXiv:1301.0664 [math.MG], 2013.
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, Discrete Comput. Geom. 52 (2014), no. 4, 754--779. MR3279548.
Brian Galebach, Tiling 132 (in list of 4-uniform tilings).
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
N. J. A. Sloane, A portion of the 3-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}. The four black dots labeled P,Q,R,S show the four types of point. The present sequence is for a point of type S.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250123, A250124, A250126.

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).

Empirical g.f.: -(x^17 +x^16 +x^15 +x^14 -2*x^13 -4*x^12 -6*x^11 -7*x^10 -11*x^9 -18*x^8 -16*x^7 -19*x^6 -14*x^5 -13*x^4 -11*x^3 -6*x^2 -4*x -1) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Dec 02 2014

Galebach link from Joseph Myers, Nov 30 2014

Extended by Joseph Myers, Dec 02 2014

A250126 Coordination sequence of point of type 3.3.4.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Original entry on oeis.org

1, 4, 9, 9, 12, 19, 21, 28, 27, 31, 38, 40, 48, 44, 49, 56, 57, 67, 63, 69, 73, 75, 85, 80, 88, 92, 95, 102, 98, 106, 109, 114, 121, 118, 123, 127, 132, 138, 137, 142, 147, 149, 156, 155, 159, 166, 168, 176, 172, 177, 184, 185, 195, 191, 197, 201, 203, 213, 208

Offset: 0

N. J. A. Sloane, Nov 29 2014

nonn

This tiling appears as an example in Connelly et al. (2014), Fig. 6 (the heavy black lines in the figures here are an artifact from that figure).

For the definition of k-uniform tiling see Section 2.2 of Chapter 2 of Grünbaum and Shephard (1987).

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.

Joseph Myers, Table of n, a(n) for n = 0..1000
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, arXiv:1301.0664 [math.MG], 2013.
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith, Ball Packings with Periodic Constraints, Discrete Comput. Geom. 52 (2014), no. 4, 754--779. MR3279548.
Brian Galebach, Tiling 132 (in list of 4-uniform tilings).
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
N. J. A. Sloane, A portion of the 3-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}. The four black dots labeled P,Q,R,S show the four types of point. The present sequence is for a point of type Q.
N. J. A. Sloane, Shows layers a(0)-a(6)

Cf. A250123, A250124, A250125.

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).

Empirical g.f.: -(2*x^16 +x^14 -2*x^12 -7*x^11 -10*x^10 -10*x^9 -14*x^8 -18*x^7 -17*x^6 -18*x^5 -12*x^4 -9*x^3 -9*x^2 -4*x -1) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)*(x^6 +x^5 +x^4 +x^3 +x^2 +x +1)). - Colin Barker, Dec 02 2014

Galebach link from Joseph Myers, Nov 30 2014

Extended by Joseph Myers, Dec 02 2014

A322038 Irregular triangle read by rows: for n >= 0, row n gives the coordination sequence for the tiling of a flat torus by a square grid with n points along each circuit.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 4, 1, 4, 6, 4, 1, 1, 4, 8, 8, 4, 1, 4, 8, 10, 8, 4, 1, 1, 4, 8, 12, 12, 8, 4, 1, 4, 8, 12, 14, 12, 8, 4, 1, 1, 4, 8, 12, 16, 16, 12, 8, 4, 1, 4, 8, 12, 16, 18, 16, 12, 8, 4, 1, 1, 4, 8, 12, 16, 20, 20, 16, 12, 8, 4

Offset: 0

N. J. A. Sloane, Dec 01 2018

More precisely, this is the coordination sequence for the quotient graph Z^2 / (nZ X nZ). The graph has n^2 vertices.

There are obvious generalizations: for example, Z^2 / (mZ X nZ) where m and n are not necessarily equal.

			The triangle begins:
1,
1,
1, 2, 1,
1, 4, 4,
1, 4, 6, 4, 1,
1, 4, 8, 8, 4,
1, 4, 8, 10, 8, 4, 1,
1, 4, 8, 12, 12, 8, 4,
1, 4, 8, 12, 14, 12, 8, 4, 1,
1, 4, 8, 12, 16, 16, 12, 8, 4,
1, 4, 8, 12, 16, 18, 16, 12, 8, 4, 1,
...

N. J. A. Sloane, Illustration showing the tilings and coordination sequences for n = 4 and 5

The rows converge to A008574.

Since the underlying graphs are finite, the coordination sequences are polynomial P_n(x).
For n even, P_n(x) = (1+x)^2*(Sum_{i=0..(n-2)/2} x^i)^2;
for n odd, P_n(x) = (1 + 2*Sum_{i=0..(n-1)/2} x^i)^2.

A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 4, 1, 6, 18, 1, 8, 32, 88, 1, 10, 50, 170, 450, 1, 12, 72, 292, 912, 2364, 1, 14, 98, 462, 1666, 4942, 12642, 1, 16, 128, 688, 2816, 9424, 27008, 68464, 1, 18, 162, 978, 4482, 16722, 53154, 148626, 374274, 1, 20, 200, 1340, 6800, 28004, 97880, 299660, 822560, 2060980, 1, 22, 242, 1782, 9922, 44726, 170610, 568150, 1690370, 4573910, 11414898

Offset: 0

R. J. Mathar, Apr 21 2021

			The full array starts
     1      2      2      2      2      2      2      2      2
     1      4      8     12     16     20     24     28     32
     1      6     18     38     66    102    146    198    258
     1      8     32     88    192    360    608    952   1408
     1     10     50    170    450   1002   1970   3530   5890
     1     12     72    292    912   2364   5336  10836  20256
     1     14     98    462   1666   4942  12642  28814  59906
     1     16    128    688   2816   9424  27008  68464 157184
     1     18    162    978   4482  16722  53154 148626 374274

J. Schroder, Generalized Schroder Numbers and the Rotation principle, J. Int. Seq. 10 (2007) # 07.7.7, Theorem 4.2.

Cf. A035607 (by antidiags), A008574 (n=1), A005899 (n=2), A008412 (n=3), A008413 (n=4), A008414 (n=5), A001105 (k=2), A035597 (k=3), A035598 (k=4).

Main diagonal gives A050146(n+1).

A343599 := proc(n,k)
    local g,x,y ;
    g := (1+y)/(1-x-y-x*y) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,k) ;
end proc:

T[n_, k_] := Module[{x, y}, SeriesCoefficient[(1 + y)/(1 - x - y - x*y), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 16 2023 *)

G.f.: (1+y)/(1-x-y-x*y).

T(n,k) = A008288(n,k) + A008288(n,k-1).

A058394 A square array based on natural numbers (A000027) with each term being the sum of 2 consecutive terms in the previous row.

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 2, 2, 1, 3, 2, 3, 3, 1, 0, 3, 4, 5, 4, 1, 4, 3, 5, 7, 8, 5, 1, 0, 4, 6, 9, 12, 12, 6, 1, 5, 4, 7, 11, 16, 20, 17, 7, 1, 0, 5, 8, 13, 20, 28, 32, 23, 8, 1, 6, 5, 9, 15, 24, 36, 48, 49, 30, 9, 1, 0, 6, 10, 17, 28, 44, 64, 80, 72, 38, 10, 1, 7, 6, 11, 19, 32, 52, 80, 112, 129

Offset: 0

Henry Bottomley, Nov 24 2000

Changing the formula by replacing T(2n,0)=T(n,2) by T(2n,0)=T(n,m) for some other value of m, would make the generating function change to coefficient of x^n in expansion of (1+x)^k/(1-x^2)^m. This would produce A058393, A058395, A057884 (and effectively A007318).

			Rows are (1,0,2,0,3,0,4,...), (1,1,2,2,3,3,...), (1,2,3,4,5,6,...), (1,3,5,7,9,11,...), etc.

Rows are A027656 (A000027 with zeros), A008619, A000027, A005408, A008574 etc. Columns are A000012, A001477, A022856 etc. Diagonals include A034007, A045891, A045623, A001792, A001787, A000337, A045618, A045889, A034009, A055250, A055251 etc. The triangle A055249 also appears in half of the array.

T(n, k)=T(n-1, k-1)+T(n, k-1) with T(0, k)=1, T(2n, 0)=T(n, 2) and T(2n+1, 0)=0. Coefficient of x^n in expansion of (1+x)^k/(1-x^2)^2.

A179000 Array T(n,k) read by antidiagonals: coefficient [x^k] of (1 + n*Sum_{i>=1} x^i)^2, k >= 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 8, 4, 1, 8, 15, 12, 5, 1, 10, 24, 24, 16, 6, 1, 12, 35, 40, 33, 20, 7, 1, 14, 48, 60, 56, 42, 24, 8, 1, 16, 63, 84, 85, 72, 51, 28, 9, 1, 18, 80, 112, 120, 110, 88, 60, 32, 10

Offset: 1

Gary W. Adamson, Jan 03 2011

Antidiagonal sums are in A136396.

			First few rows of the array:
  1   2   3   4   5   6   7   8   9  10  11  A000027
  1   4   8  12  16  20  24  28  32  36  40  A008574
  1   6  15  24  33  42  51  60  69  78  87  A122709
  1   8  24  40  56  72  88 104 120 136 152  A051062
  1  10  35  60  85 110 135 160 185 210 235
  1  12  48  84 120 156 192 228 264 300 336
  1  14  63 112 161 210 259 308 357 406 455
  1  16  80 144 208 272 336 400 464 528 592
  1  18  99 180 261 342 423 504 585 666 747
Row n=3 is generated by (1 + 3x + 3x^2 + 3x^3 + 3x^4 + ...)^2 = 1 + 6x + 15x^2 + 24x^3 + ..., for example.

Cf. A136396, A179901.

A179000 := proc(n,k) if k = 0 then 1; else 2*n+n^2*(k-1) ; end if; end proc: # R. J. Mathar, Jan 05 2011

T(n,0) = 1; T(n,k) = n*(2+n*(k-1)), k > 0. - R. J. Mathar, Jan 05 2011

cp's OEIS Frontend

A130713 a(0)=a(2)=1, a(1)=2, a(n)=0 for n > 2.

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Maple

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A241204 Expansion of (1 + 2*x)^2/(1 - 2*x)^2.

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Magma

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A250123 Coordination sequence of point of type 3.3.4.3.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

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A250124 Coordination sequence of point of type 3.3.12.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

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A250125 Coordination sequence of point of type 3.4.3.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

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A250126 Coordination sequence of point of type 3.3.4.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

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A322038 Irregular triangle read by rows: for n >= 0, row n gives the coordination sequence for the tiling of a flat torus by a square grid with n points along each circuit.

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A343599 T(n,k) is the coordination number of the (n+1)-dimensional cubic lattice for radius k; triangle read by rows, n>=0, 0<=k<=n.

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A241204 Expansion of (1 + 2x)^2/(1 - 2x)^2.