A001018 -id:A001018 - cp's OEIS frontend

A165426 a(1) = 1, a(2) = 8, a(n) = product of the previous terms for n >= 3.

1, 8, 8, 64, 4096, 16777216, 281474976710656, 79228162514264337593543950336, 6277101735386680763835789423207666416102355444464034512896

Offset: 1

Jaroslav Krizek, Sep 17 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 1..13

a[1]:= 1; a[2]:= 8; a[n_]:= Product[a[j], {j,1,n-1}]; Table[a[n],{n,1, 12}] (* G. C. Greubel, Oct 19 2018 *)

{a(n) = if(n==1, 1, if(n==2, 8, prod(j=1,n-1, a(j))))};
for(n=1,10, print1(a(n), ", ")) \\ G. C. Greubel, Oct 19 2018

a(1) = 1, a(2) = 8, a(n) = Product_{i=1..n-1} a(i), n >= 3.
a(1) = 1, a(2) = 8, a(n) = A001018(2^(n-3)) = 8^(2^(n-3)), n >= 3.
a(1) = 1, a(2) = 8, a(3) = 8, a(n) = (a(n-1))^2, n >= 4.
a(n) = 8^A166444(n). [uncovered by sequencedb.net]. - R. J. Mathar, Jun 30 2021

A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7

Offset: 0

Alois P. Heinz, Jan 15 2021

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,      0,       0,        0, ...
  1, 1,  1,   1,    1,     1,      1,       1,        1, ...
  1, 2,  4,   8,   16,    32,     64,     128,      256, ...
  3, 3,  3,   3,    3,     3,      3,       3,        3, ...
  1, 4, 16,  64,  256,  1024,   4096,   16384,    65536, ...
  3, 5,  9,  17,   33,    65,    129,     257,      513, ...
  3, 6, 12,  24,   48,    96,    192,     384,      768, ...
  7, 7,  7,   7,    7,     7,      7,       7,        7, ...
  1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened

Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
Rows n=0..17, 19 give: A000004, A000012, A000079, A010701, A000302, A000051(k+1), A007283, A010727, A001018, A087289, A007582(k+1), A062709(k+2), A164346, A181565(k+1), A005009, A181404(k+3), A001025, A199493, A253208(k+1).
Main diagonal gives A340667.
Cf. A000120, A023416.

A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n<2, n,
     `if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)

A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).

A023416(A(n,k)) = k * A023416(n) for n >= 1.

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

A001555 a(n) = 1^n + 2^n + ... + 8^n.

Original entry on oeis.org

8, 36, 204, 1296, 8772, 61776, 446964, 3297456, 24684612, 186884496, 1427557524, 10983260016, 84998999652, 660994932816, 5161010498484, 40433724284976, 317685943157892, 2502137235710736, 19748255868485844, 156142792528260336, 1236466399775623332

Offset: 0

N. J. A. Sloane

Conjectures for o.g.f.s for this type of sequence appear in the PhD thesis by Simon Plouffe. See A001552 for the reference. These conjectures are proved in a link given in A196837. [Wolfdieter Lang, Oct 15 2011]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
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 and Robert Israel, Table of n, a(n) for n = 0..1000 (n = 0..200 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 368
Index entries for linear recurrences with constant coefficients, signature (36, -546, 4536, -22449, 67284, -118124, 109584, -40320).

Column 8 of array A103438.

Cf. A001018, A001552, A001554, A196837.

seq(add(j^n,j=1..8), n=0..20); # Robert Israel, Aug 23 2015

Table[Total[Range[8]^n], {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)

first(m)=vector(m,n,n--;sum(i=1,8,i^n)) \\ Anders Hellström, Aug 23 2015

From Wolfdieter Lang, Oct 15 2011 (Start)
E.g.f.: (1-exp(8*x))/(exp(-x)-1) = Sum_{j=1..8} exp(j*x) (trivial).
O.g.f.: 4*(2-9*x)*(1-27*x+288*x^2-1539*x^3+4299*x^4-5886*x^5+3044*x^6) / Product_{j=1..8} (1-j*x). From the e.g.f. via Laplace transformation. See the proof in a link under A196837. (End)
a(n) = A001554(n) + A001018(n). - Michel Marcus, Jul 26 2013

More terms from Jon E. Schoenfield, Mar 24 2010

A013614 Triangle of coefficients in expansion of (1+7x)^n.

Original entry on oeis.org

1, 1, 7, 1, 14, 49, 1, 21, 147, 343, 1, 28, 294, 1372, 2401, 1, 35, 490, 3430, 12005, 16807, 1, 42, 735, 6860, 36015, 100842, 117649, 1, 49, 1029, 12005, 84035, 352947, 823543, 823543, 1, 56, 1372, 19208, 168070, 941192, 3294172, 6588344, 5764801

Offset: 0

N. J. A. Sloane

T(n,k) equals the number of n-length words on {0,1,...,7} having n-k zeros. - Milan Janjic, Jul 24 2015

			Triangle starts:
1;
1, 7;
1, 14, 49;
1, 21, 147, 343;
1, 28, 294, 1372, 2401;
1, 35, 490, 3430, 12005, 16807;
...

Cf. A000420 (right edge).

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+7*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 24 2015

T[n_, k_] := 7^k*Binomial[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 23 2016 *)

G.f.: 1 / (1 - x(1+7y)).

T(n,k) = 7^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*6^(n-i). Row sums are 8^n = A001018. - Mircea Merca, Apr 28 2012

A038484 Sums of 2 distinct powers of 8.

Original entry on oeis.org

9, 65, 72, 513, 520, 576, 4097, 4104, 4160, 4608, 32769, 32776, 32832, 33280, 36864, 262145, 262152, 262208, 262656, 266240, 294912, 2097153, 2097160, 2097216, 2097664, 2101248, 2129920, 2359296, 16777217, 16777224, 16777280, 16777728, 16781312, 16809984, 17039360, 18874368

Offset: 1

Olivier Gérard

Amiram Eldar, Table of n, a(n) for n = 1..10000

Base-8 interpretation of A038444.

Cf. A001018, A038485, A038486.

Total/@Subsets[8^Range[0,10],{2}]//Union (* Harvey P. Dale, Jul 04 2022 *)

from math import isqrt
def A038484(n): return (1<<(a:=isqrt(n<<3)+1&-2)+(m:=a>>1))+(1<<3*(n-1-(m*(m-1)>>1))) # Chai Wah Wu, Apr 04 2025

More terms from Vincenzo Librandi, Aug 06 2009

Offset corrected by Amiram Eldar, Jul 14 2022

cp's OEIS Frontend

A165426 a(1) = 1, a(2) = 8, a(n) = product of the previous terms for n >= 3.

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A340666 A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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