A365606 -id:A365606 - cp's OEIS frontend

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

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

A367702 Number of degree 4 vertices in the n-Menger sponge graph.

Original entry on oeis.org

0, 144, 2784, 57552, 1180320, 23889936, 480221280, 9624275280, 192645717024, 3854200280208, 77094305873376, 1541968557881808, 30840030795738528, 616805893363960080, 12336160087905835872, 246723539526229152336, 4934473492678780614432, 98689491470837087102352

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) = 0.

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.
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},{0,144,2784,57552},25] (* Paolo Xausa, Nov 29 2023 *)

def A367702(n): return ((5**n<<(n<<1)+5)-(17<<(3*n+2))+(3**(n+4)<<3))//85-24 # Chai Wah Wu, Nov 28 2023

a(n) = (32/85)*20^n - (4/5)*8^n + (648/85)*3^n - 24.
a(n) = 20*a(n-1) + (6/5)*8^n - (216/5)*3^n + 456.
a(n) = 20^n - A367700(n) - A367701(n) - A367706(n) - A367707(n).
4*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 5*A367706(n) - 6*A367707(n).
G.f.: 12*x^2*(7 - 224*x + 1865*x^2 - 4308*x^3)/(5*(1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 28 2023

A367706 Number of degree 5 vertices in the n-Menger sponge graph.

Original entry on oeis.org

0, 24, 1272, 27192, 537720, 10638648, 211640184, 4223114808, 84382898808, 1687017131832, 33735198879096, 674662776506424, 13492925768472696, 269855876817045816, 5397096426544159608, 107941759648376656440, 2158833841895083390584, 43176666029284877542200, 863533234116651651590520

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) = 0.

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.
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},{0,24,1272,27192},25] (* Paolo Xausa, Nov 29 2023 *)

def A367706(n): return ((7*5**n<<(n<<1)+1)+(17<<(3*n+1))-(3**(n+3)<<5))//85+24 # Chai Wah Wu, Nov 28 2023

a(n) = (14/85)*20^n + (2/5)*8^n - (864/85)*3^n + 24.
a(n) = 20*a(n-1) - (3/5)*8^n + (288/5)*3^n - 456.
a(n) = 20^n - A367700(n) - A367701(n) - A367702(n) - A367707(n).
5*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 4*A365602(n) - 6*A367707(n).
G.f.: 24*x^2*(1 + 21*x - 288*x^2)/((1 - x)*(1- 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 28 2023

A367707 Number of degree 6 vertices in the n-Menger sponge graph.

Original entry on oeis.org

0, 8, 456, 14312, 338376, 7218536, 148082760, 2991665384, 60074332872, 1203417692264, 24083810625864, 481799892270056, 9636987359949768, 192747663544965992, 3855016602355831368, 77100838700834961128, 1542020827252644619464, 30840448970959051746920, 616809238826486098348872

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) = 0.

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.
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},{0,8,456,14312},25] (* Paolo Xausa, Nov 29 2023 *)

def A367707(n): return ((5**(n+1)<<(n<<1)+1)-(51<<(3*n+1))+(3**(n+3)<<4))//85-8 # Chai Wah Wu, Nov 28 2023

a(n) = (2/17)*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) - A367701(n) - A367702(n) - A367706(n).
6*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 4*A365602(n) - 5*A367706(n).
G.f.: 8*x^2*(1 + 25*x + 240*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 28 2023

A291775 Domination number of the n-Sierpinski carpet graph.

Original entry on oeis.org

3, 18, 130, 1026

Offset: 1

Eric W. Weisstein, Aug 31 2017

Also the lower independence number (=independent domination number) of the n-Sierpinski carpet graph. - Eric W. Weisstein, Aug 02 2023
From Allan Bickle, Aug 10 2024: (Start)
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.
Conjecture: For n>1, a(n) = 2^(3n-2) + 2.  There is an independent dominating set of this size consisting of the vertices on every third diagonal and two corner vertices.
(End)

			The 8-cycle has domination number 3, so a(1) = 3.

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, Domination Number
Eric Weisstein's World of Mathematics, Lower Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Carpet Graph

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A292707, A347651 (vertex sets).

A381517 Perimeter of the Sierpiński carpet at iteration n.

Original entry on oeis.org

4, 16, 80, 496, 3536, 26992, 212048, 1684720, 13442768, 107437168, 859182416, 6872514544, 54977282000, 439809752944, 3518452514384, 28147543587568, 225180119118032, 1801440264196720, 14411520047331152, 115292154179921392, 922337214843187664, 7378697662956950896, 59029581136289955920, 472236648588222693616

Offset: 0

Jakub Buczak, Feb 26 2025

Carpet n has an overall size 3^n X 3^n and the perimeter here includes the perimeter of all holes within it.
Carpet n=0 is a unit square and has perimeter a(0) = 4.
Carpet n can be constructed by arranging 8 copies of carpet n-1 in a square with a hole in the middle,
  X  X  X
  X     X
  X  X  X
There are no gaps in each side so 2 sides of each n-1 are now not on the perimeter so a(n) = 8*a(n-1) - 16*3^(n-1).
An equivalent construction is to replace each of the 8^(n-1) unit squares of carpet n-1 with a 3 X 3 block of unit squares with a hole in the middle, so that a(n) = 3*a(n-1) + 4*8^(n-1).
A fractal is obtained by scaling the whole carpet down to a unit square and its scaled perimeter a(n)/3^n -> oo shows the perimeter is infinite even though the area is bounded.

			For n=0, a(0) = 4, the geometric representation is a square.
For n=3, a(3) = 496.

Jakub Buczak, Perimeter of the Sierpiński Carpet
Michael Small, Brendan Florio, and Phillip Donald Fawell, The use of the perimeter area method to calculate the fractal dimension of aggregates, see section 3.2 equation (27) where a(n) = P_s(n+1) with scale factor g_1 = 1.
Eric Weisstein's World of Mathematics, Sierpiński Carpet

Cf. A001018, A271939, A293143, A365606.

Cf. A113210 (fractal dimension).

a = lambda n: (4 * (4 * 3**n + 8**n)) // 5

a(n) = (4/5)*(4*3^n + 8^n).

a(n) = A365606(n+1) - 4.

cp's OEIS Frontend

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

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

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

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

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A367702 Number of degree 4 vertices in the n-Menger sponge graph.

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A367706 Number of degree 5 vertices in the n-Menger sponge graph.

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A367707 Number of degree 6 vertices in the n-Menger sponge graph.

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