A291066 -id:A291066 - cp's OEIS frontend

A009964 Powers of 20.

1, 20, 400, 8000, 160000, 3200000, 64000000, 1280000000, 25600000000, 512000000000, 10240000000000, 204800000000000, 4096000000000000, 81920000000000000, 1638400000000000000, 32768000000000000000

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 20), L(1, 20), P(1, 20), T(1, 20). Essentially same as Pisot sequences E(20, 400), L(20, 400), P(20, 400), T(20, 400). See A008776 for definitions of Pisot sequences.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 20-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) gives the number of small cubes in the n-th iteration of the Menger sponge fractal. - Felix Fröhlich, Jul 09 2016
Equivalently, the number of vertices in the n-Menger sponge graph.

T. D. Noe, Table of n, a(n) for n = 0..100
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Menger Sponge
Eric Weisstein's World of Mathematics, Menger Sponge Graph
Eric Weisstein's World of Mathematics, Vertex Count
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (20).

Cf. A291066 (edge count).
Cf. A000079, A011557; A000302, A000351; A000244, A159991.
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

List([0..20],n->20^n); # Muniru A Asiru, Nov 21 2018

[20^n: n in [0..100]] // Vincenzo Librandi, Nov 21 2010

[20^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

20^Range[0, 10] (* or *) LinearRecurrence[{20}, {1}, 20] (* Eric W. Weisstein, Aug 17 2017 *)

makelist(20^n,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=20^n \\ Charles R Greathouse IV, Jun 19 2015

powers(20,12) \\ Charles R Greathouse IV, Jun 19 2015

[20**n for n in range(21)] # Stefano Spezia, Nov 21 2018

[20^n for n in range(21)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-20*x).
E.g.f.: exp(20*x).
a(n) = A159991(n)/A000244(n). - Reinhard Zumkeller, May 02 2009
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 20^n.
a(n) = 20*a(n-1) for n > 0, a(0) = 1. (End)
a(n) = A000079(n)*A011557(n) = A000302(n)*A000351(n). - Felix Fröhlich, Jul 09 2016

A083233 a(n) = (3*8^n + 0^n)/4.

Original entry on oeis.org

1, 6, 48, 384, 3072, 24576, 196608, 1572864, 12582912, 100663296, 805306368, 6442450944, 51539607552, 412316860416, 3298534883328, 26388279066624, 211106232532992, 1688849860263936, 13510798882111488, 108086391056891904, 864691128455135232

Offset: 0

Paul Barry, Apr 23 2003

Binomial transform of A083232. Inverse binomial transform of A066443.
Numbers k such that, except for some first term, k^2 = [A000302]^3 + [A004171]^3 + [A002001]^3; e.g., 3072^2 = 64^3 + 128^3 + 192^3; 51539607552^2 = 4194304^3 + 8388608^3 + 12582912^3. - Vincenzo Librandi, Aug 08 2010
With the exception of the first term, these numbers cannot be written as the sum of two integer cubes but can be written as the sum of two positive rational cubes (i.e., 6*8^n = (17*2^n/21)^3 + (37*2^n/21)^3). - Arkadiusz Wesolowski, Aug 15 2013
a(n+1) is the number of unit square faces on the convex hull of a level n Menger sponge. This follows since it has six exterior faces, each of which is a Sierpinski carpet with 8^n squares. - Allan Bickle, Nov 28 2022

			a(0) = (3*8^0 + 0^0)/4 = 4/4 = 1 (using 0^0 = 1).

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A083234. Subsequence of A159843.
Cf. A000302, A004171, A002001.
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

Join[{1},NestList[8#&,6,20]] (* Harvey P. Dale, Sep 25 2020 *)

a(n)=(3*8^n+0^n)/4 \\ Charles R Greathouse IV, Oct 07 2015

def A083233(n): return 3<<3*n-2 if n else 1 # Chai Wah Wu, Nov 27 2023

a(n) = (3*8^n + 0^n)/4.
G.f.: (1-2x)/(1-8x).
E.g.f.: (3*exp(8x) + exp(0))/4.
a(0) = 1, a(n+1) = 6*8^n. - Arkadiusz Wesolowski, Aug 15 2013

A332705 Number of unit square faces (or surface area) of a stage-n Menger sponge.

Original entry on oeis.org

6, 72, 1056, 18048, 336384, 6531072, 129048576, 2568388608, 51267108864, 1024536870912, 20484294967296, 409634359738368, 8192274877906944, 163842199023255552, 3276817592186044416, 65536140737488355328, 1310721125899906842624

Offset: 0

Eric Andres, Feb 20 2020

The values are established based on the following observation: A stage-0 Menger sponge has 6 faces (a cube). Note that a face here corresponds to the unit face of a unit cube used to construct the Menger sponge. This means that counting the faces is equivalent to computing the surface area. After that, a stage-n Menger sponge is created by replacing each of the 20 cubes of the stage-1 Menger sponge with a stage-(n-1) Menger sponge. Each of the 8 stage-(n-1) sponges on the corner loses 3 sides of outer faces (which represents a total of 8^(n-1) faces). Each of the 12 stage-(n-1) Menger sponges on the edges (between the corners) lose two sides of outer faces. Thus the recurrence formula given below.

			a(0)=6 is the number of faces of a cube.
a(1)=72 is the number of faces of a stage-1 Menger sponge, which has 6*8 faces on its convex hull, and 6*4 faces not on its convex hull.

Colin Barker, Table of n, a(n) for n = 0..750
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge Graph
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (28,-160).

Related to A135918 (Genus of stage-n Menger sponge). The ratio of this sequence / genus of the stage-n Menger sponge tends toward 38/3.
Cf. A009964 (vertices of graph), A291066 (edges of graph).
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

seq[n_] := 20 seq[n - 1] - 3*2^(4 + 3 (n - 1)) /; (n >= 1); seq[0] := 6;

Vec(6*(1 - 16*x) / ((1 - 8*x)*(1 - 20*x)) + O(x^20)) \\ Colin Barker, Feb 20 2020

def A332705(n): return (5**n+(1<Chai Wah Wu, Nov 27 2023

a(n) = 20*a(n-1) - 3*2^(1 + 3*n); with a(0)=6.
a(n) = 2^(1 + 2*n) (2^(1 + n) + 5^n) (Direct formula based on suggestion by Rémy Sigrist).
From Colin Barker, Feb 20 2020: (Start)
G.f.: 6*(1 - 16*x) / ((1 - 8*x)*(1 - 20*x)).
a(n) = 28*a(n-1) - 160*a(n-2) for n > 2. (End)
E.g.f.: 2*exp(8*x)*(2 + exp(12*x)). - Stefano Spezia, Feb 20 2020
From Allan Bickle, Nov 28 2022: (Start)
a(n) = 2*20^n + 4*8^n.
a(n) = A291066(n) + A083233(n+1). (End)

A359452 Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.

Original entry on oeis.org

1, 8, 208, 3968, 80128, 1599488, 32002048, 639991808, 12800032768, 255999868928, 5120000524288, 102399997902848, 2048000008388608, 40959999966445568, 819200000134217728, 16383999999463129088, 327680000002147483648, 6553599999991410065408, 131072000000034359738368

Offset: 0

Allan Bickle, Jan 02 2023

This sequence and the sequence counting the non-corner vertices (A359453) alternate as to which is larger.

			The level 1 Menger sponge graph can be formed by subdividing every edge of a cube graph.  This produces a graph with 8 corner vertices and 12 non-corner vertices, so a(1) = 8.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (16,80).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359453 (number of non-corner vertices).
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
Cf. A262710.

A359452[n_]:=(20^n+(-4)^n)/2;Array[A359452,25,0] (* Paolo Xausa, Nov 29 2023 *)

a(n) = (20^n + (-4)^n)/2 \\ Andrew Howroyd, Jan 02 2023

def A359452(n): return (10**n<Chai Wah Wu, Feb 13 2023

a(n) = (20^n + (-4)^n)/2.
a(n) = (A009964(n) + A262710(n))/2.
a(n) = 20^n - A359453(n).
From Stefano Spezia, Jan 02 2023: (Start)
O.g.f.: (1 - 8*x)/((1 - 20*x)*(1 + 4*x)).
E.g.f.: exp(8*x)*cosh(12*x). (End)

A359453 Number of vertices in the partite set of the n-Menger sponge graph that do not contain the corners.

Original entry on oeis.org

0, 12, 192, 4032, 79872, 1600512, 31997952, 640008192, 12799967232, 256000131072, 5119999475712, 102400002097152, 2047999991611392, 40960000033554432, 819199999865782272, 16384000000536870912, 327679999997852516352, 6553600000008589934592, 131071999999965640261632

Offset: 0

Allan Bickle, Jan 02 2023

This sequence and the sequence counting the corner vertices (A359452) alternate as to which is larger.

			The level 1 Menger sponge graph can be formed by subdividing every edge of a cube graph.  This produces a graph with 8 corner vertices and 12 non-corner vertices, so a(1) = 12.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (16,80).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359452 (number of corner vertices).
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

A359453[n_]:=(20^n-(-4)^n)/2;Array[A359453,25,0] (* Paolo Xausa, Nov 30 2023 *)

a(n) = (20^n - (-4)^n)/2 \\ Andrew Howroyd, Jan 02 2023

def A359453(n): return (10**n<Chai Wah Wu, Feb 13 2023

a(n) = (20^n - (-4)^n)/2.
a(n) = (A009964(n) - A262710(n))/2.
a(n) = 20^n - A359452(n).
From Stefano Spezia, Jan 02 2023: (Start)
O.g.f.: 12*x/((1 - 20*x)*(1 + 4*x)).
E.g.f.: (cosh(8*x) + sinh(8*x))*sinh(12*x). (End)

A365606 Number of degree 2 vertices in the n-Sierpinski carpet graph.

Original entry on oeis.org

8, 20, 84, 500, 3540, 26996, 212052, 1684724, 13442772, 107437172, 859182420, 6872514548, 54977282004, 439809752948, 3518452514388, 28147543587572, 225180119118036, 1801440264196724, 14411520047331156, 115292154179921396, 922337214843187668, 7378697662956950900, 59029581136289955924

Offset: 1

Allan Bickle, Sep 12 2023

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 8.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

LinearRecurrence[{12,-35,24},{8,20,84},30] (* Paolo Xausa, Oct 16 2023 *)

def A365606(n): return ((1<<3*n-1)+(3**(n-1)<<4))//5+4 # Chai Wah Wu, Nov 27 2023

a(n) = (1/10)*8^n + (16/15)*3^n + 4.
a(n) = 8*a(n-1) - 16*3^(n-2) - 28.
a(n) = 8^n - A365607(n) - A365608(n).
2*a(n) = 2*A271939(n) - 3*A365607(n) - 4*A365608(n).
G.f.: 4*x*(2 - 19*x + 31*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023

A365607 Number of degree 3 vertices in the n-Sierpinski carpet graph.

Original entry on oeis.org

0, 40, 328, 2536, 19912, 158056, 1260616, 10073320, 80551624, 644308072, 5154149704, 41232252904, 329855188936, 2638833008488, 21110638558792, 168885031942888, 1351080025960648, 10808639518937704, 86469114085259080, 691752906483344872, 5534023233270575560, 44272185810376054120

Offset: 1

Allan Bickle, Sep 12 2023

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 0.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

LinearRecurrence[{12,-35,24},{0,40,328},30] (* Paolo Xausa, Oct 16 2023 *)

def A365607(n): return ((3<<3*n)+(3**(n-1)<<4))//5-8 # Chai Wah Wu, Nov 27 2023

a(n) = (3/5)*8^n + (16/15)*3^n - 8.
a(n) = 8*a(n-1) - 16*3^(n-2) + 56.
a(n) = 8^n - A365606(n) - A365608(n).
3*a(n) = 2*A271939(n) - 2*A365606(n) - 4*A365608(n).
G.f.: 8*x^2*(5 - 19*x)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023

A365608 Number of degree 4 vertices in the n-Sierpinski carpet graph.

Original entry on oeis.org

0, 4, 100, 1060, 9316, 77092, 624484, 5019172, 40223332, 321996580, 2576602468, 20614709284, 164923342948, 1319403749668, 10555281015652, 84442401180196, 675539668606564, 5404318726347556, 43234553943265636, 345876443943580708, 2767011588741012580, 22136092821505201444, 177088742906772914020

Offset: 1

Allan Bickle, Sep 12 2023

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 0.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

LinearRecurrence[{12,-35,24},{0,4,100},30] (* Paolo Xausa, Oct 16 2023 *)

def A365608(n): return ((3<<3*n-1)-(3**(n-1)<<5))//5+4 # Chai Wah Wu, Nov 27 2023

a(n) = (3/10)*8^n - (32/15)*3^n + 4.
a(n) = 8*a(n-1) + 32*3^(n-2) - 28.
a(n) = 8^n - A365606(n) - A365607(n).
4*a(n) = 2*A271939(n) - 2*A365606(n) - 3*A365607(n).
G.f.: 4*x^2*(1 + 13*x)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023

A367700 Number of degree 2 vertices in the n-Menger sponge graph.

Original entry on oeis.org

12, 72, 744, 11256, 201960, 3871416, 76138536, 1512609912, 30171384168, 602782587960, 12050495247528, 240968665611768, 4819043435788776, 96378229818994104, 1927543485550004520, 38550700825394191224, 771012665426135994984, 15420242499878035355448, 308404763528431125030312

Offset: 1

Allan Bickle, Nov 27 2023

The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.

			The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices.  Thus a(1) = 12.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge Graph.
Index entries for linear recurrences with constant coefficients, signature (31,-244,480).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359452, A359453 (numbers of corner and non-corner vertices).
Cf. A083233, A332705 (surface area).
Cf. A367701, A367702, A367706, A367707 (degrees 2 through 6).
Cf. A001018, A271939, A365602, A365606, A365607, A365608 (Sierpinski carpet graphs).

LinearRecurrence[{31,-244,480}, {12, 72, 744}, 25] (* Paolo Xausa, Nov 28 2023 *)

def A367700(n): return (5*20**n+(34<<3*n)+216*3**n)//85 # Chai Wah Wu, Nov 27 2023

a(n) = (1/17)*20^n + (2/5)*8^n + (216/85)*3^n.
a(n) = 20*a(n-1) - (3/5)*8^n - (72/5)*3^n.
a(n) = 20^n - A367701(n) - A367702(n) - A367706(n) - A367707(n).
2*a(n) = 2*A291066(n) - 3*A367701(n) - 4*A365602(n) - 5*A367706(n) - 6*A367707(n).
G.f.: 12*x*(1 - 25*x + 120*x^2)/((1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 27 2023

A367701 Number of degree 3 vertices in the n-Menger sponge graph.

Original entry on oeis.org

8, 152, 2744, 49688, 941624, 18381464, 363917240, 7248334616, 144725667128, 2892582307736, 57836189374136, 1156600107729944, 23131012640050232, 462612336455034008, 9252183397644168632, 185043161299165038872, 3700859172747355380536, 74017151029040948253080

Offset: 1

Allan Bickle, Nov 27 2023

The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.

			The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices.  Thus a(1) = 8.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge Graph.
Index entries for linear recurrences with constant coefficients, signature (32,-275,724,-480).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359452, A359453 (numbers of corner and non-corner vertices).
Cf. A291066, A083233, A332705 (surface area).
Cf. A367700, A367701, A367702, A367706, A367707 (degrees 2 through 6).
Cf. A001018, A271939, A365606, A365607, A365608 (Sierpinski carpet graphs).

LinearRecurrence[{32,-275,724,-480},{8,152,2744,49688},25] (* Paolo Xausa, Nov 28 2023 *)

def A367701(n): return ((3*5**n<<(n<<1)+3)+(51<<(3*n+1))-(3**(n+3)<<4))//85+8 # Chai Wah Wu, Nov 28 2023

a(n) = (24/85)*20^n + (6/5)*8^n - (432/85)*3^n + 8.
a(n) = 20*a(n-1) - (9/5)*8^n + (144/5)*3^n - 152.
a(n) = 20^n - A367700(n) - A367702(n) - A367706(n) - A367707(n).
3*a(n) = 2*A291066(n) - 2*A367700(n) - 4*A365602(n) - 5*A367706(n) - 6*A367707(n).
G.f.: 8*x*(1 - 13*x + 10*x^2 - 264*x^3)/((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 27 2023

cp's OEIS Frontend

A009964 Powers of 20.

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A083233 a(n) = (3*8^n + 0^n)/4.

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A332705 Number of unit square faces (or surface area) of a stage-n Menger sponge.

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A359452 Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.

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A359453 Number of vertices in the partite set of the n-Menger sponge graph that do not contain the corners.

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A365606 Number of degree 2 vertices in the n-Sierpinski carpet graph.

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