A271939 -id:A271939 - cp's OEIS frontend

A001018 Powers of 8: a(n) = 8^n.

1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 134217728, 1073741824, 8589934592, 68719476736, 549755813888, 4398046511104, 35184372088832, 281474976710656, 2251799813685248, 18014398509481984, 144115188075855872, 1152921504606846976, 9223372036854775808, 73786976294838206464, 590295810358705651712, 4722366482869645213696

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 8), L(1, 8), P(1, 8), T(1, 8). Essentially same as Pisot sequences E(8, 64), L(8, 64), P(8, 64), T(8, 64). See A008776 for definitions of Pisot sequences.
If X_1, X_2, ..., X_n is a partition of the set {1..2n} into blocks of size 2 then, for n>=1, a(n) is equal to the number of functions f : {1..2n} -> {1,2,3} such that for fixed y_1,y_2,...,y_n in {1,2,3} we have f(X_i)<>{y_i}, (i=1..n). - Milan Janjic, May 24 2007
This is the auto-convolution (convolution square) of A059304. - R. J. Mathar, May 25 2009
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 8-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) is equal to the determinant of a 3 X 3 matrix with rows 2^(n+2), 2^(n+1), 2^n; 2^(n+3), 2^(n+4), 2(n+3); 2^n, 2^(n+1), 2^(n+2) when it is divided by 144. - J. M. Bergot, May 07 2014
a(n) gives the number of small squares in the n-th iteration of the Sierpinski carpet fractal. Equivalently, the number of vertices in the n-Sierpinski carpet graph. - Allan Bickle, Nov 27 2022

			For n=1, the 1st order Sierpinski carpet graph is an 8-cycle.

K. H. Rosen et al., eds., Handbook of Discrete and Combinatorial Mathematics, CRC Press, 2017; p. 15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 273
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Tanya Khovanova, Recursive Sequences
Caroline Nunn, A Proof of a Generalization of Niven's Theorem Using Algebraic Number Theory, Rose-Hulman Undergraduate Mathematics Journal: Vol. 22, Iss. 2, Article 3 (2021). See table at p. 9.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Sierpiński Carpet
Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A013730, A103333, A013731, A067417, A083233, A055274.
Cf. A008588, A016969, A157176.
Cf. A000079 (powers of 2), A000244 (powers of 3), A000302 (powers of 4), A000351 (powers of 5), A000400 (powers of 6), A000420 (powers of 7), A001019 (powers of 9), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48), A087752 (powers of 49), A165800 (powers of 50), A159991 (powers of 60).
Cf. A032766 (floor(3*n/2)).
Cf. A271939 (number of edges in the n-Sierpinski carpet graph).

a001018 = (8 ^)
a001018_list = iterate (* 8) 1  -- Reinhard Zumkeller, Apr 29 2015

[8^n : n in [0..30]]; // Wesley Ivan Hurt, Sep 27 2016

seq(8^n, n=0..23); # Nathaniel Johnston, Jun 26 2011
A001018 := n -> 8^n; # M. F. Hasler, Apr 19 2015

Table[8^n, {n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

makelist(8^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

a(n)=8^n \\ Charles R Greathouse IV, May 10 2014

print([8**n for n in range(25)]) # Michael S. Branicky, Dec 29 2021

a(n) = 8^n.
a(0) = 1; a(n) = 8*a(n-1) for n > 0.
G.f.: 1/(1-8*x).
E.g.f.: exp(8*x).
Sum_{n>=0} 1/a(n) = 8/7. - Gary W. Adamson, Aug 29 2008
a(n) = A157176(A008588(n)); a(n+1) = A157176(A016969(n)). - Reinhard Zumkeller, Feb 24 2009
From Stefano Spezia, Dec 28 2021: (Start)
a(n) = (-1)^n*(1 + sqrt(-3))^(3*n) (see Nunn, p. 9).
a(n) = (-1)^n*Sum_{k=0..floor(3*n/2)} (-3)^k*binomial(3*n, 2*k) (see Nunn, p. 9). (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

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).

cp's OEIS Frontend

A001018 Powers of 8: a(n) = 8^n.

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

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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.