A009964 -id:A009964 - cp's OEIS frontend

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

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

A009984 Powers of 40.

Original entry on oeis.org

1, 40, 1600, 64000, 2560000, 102400000, 4096000000, 163840000000, 6553600000000, 262144000000000, 10485760000000000, 419430400000000000, 16777216000000000000, 671088640000000000000, 26843545600000000000000, 1073741824000000000000000, 42949672960000000000000000

Offset: 0

N. J. A. Sloane, Dec 11 1996

Same as Pisot sequences E(1, 40), L(1, 40), P(1, 40), T(1, 40). Essentially same as Pisot sequences E(40, 1600), L(40, 1600), P(40, 1600), T(40, 1600). 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 40-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (40).

Cf. A000079, A000302, A008776, A009964, A011557, A259076.

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

40^Range[0, 19] (* Alonso del Arte, Sep 04 2016 *)

a(n)=40^n \\ Charles R Greathouse IV, Jun 19 2015

powers(40,10) \\ Charles R Greathouse IV, Jun 19 2015

G.f.: 1/(1 - 40*x). - Philippe Deléham, Nov 24 2008
a(n) = 40^n; a(n) = 40*a(n-1), a(0) = 1. - Vincenzo Librandi, Nov 21 2010
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: exp(40*x).
a(n) = A000079(n)*A009964(n) = A259076(n)/A000079(n). (End)

A013725 a(n) = 20^(2*n + 1).

Original entry on oeis.org

20, 8000, 3200000, 1280000000, 512000000000, 204800000000000, 81920000000000000, 32768000000000000000, 13107200000000000000000, 5242880000000000000000000, 2097152000000000000000000000, 838860800000000000000000000000, 335544320000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (400).

Bisection of A009964 (20^n).

Cf. A004171, A005408, A013708-A013724, A013726-A013729.

[20^(2*n+1): n in [0..15]]; // Vincenzo Librandi, Jun 26 2011

seq(20^(2*n+1),n=0..10); # Nathaniel Johnston, Jun 25 2011

NestList[400*# &, 20, 15] (* Paolo Xausa, Jul 12 2025 *)

a(n)=20^(2*n+1) \\ Charles R Greathouse IV, Jul 11 2016

From Philippe Deléham, Nov 28 2008: (Start)
a(n) = 400*a(n-1); a(0)=20.
G.f.: 20/(1-400*x). (End)
From Elmo R. Oliveira, Jul 10 2025: (Start)
E.g.f.: 20*exp(400*x).
a(n) = A004171(n)*A013715(n) = A009964(A005408(n)). (End)

A063012 Sum of distinct powers of 20; i.e., numbers with digits in {0,1} base 20; i.e., write n in base 2 and read as if written in base 20.

Original entry on oeis.org

0, 1, 20, 21, 400, 401, 420, 421, 8000, 8001, 8020, 8021, 8400, 8401, 8420, 8421, 160000, 160001, 160020, 160021, 160400, 160401, 160420, 160421, 168000, 168001, 168020, 168021, 168400, 168401, 168420, 168421, 3200000, 3200001, 3200020, 3200021, 3200400, 3200401

Offset: 0

Henry Bottomley, Jul 04 2001

			a(5) = 401 since 5 written in base 2 is 101 so a(5) = 1*20^2 + 0*20^1 + 1*20^0 = 400 + 0 + 1 = 401.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

A001477, A005836, A000695, A033042, A033043, A033044, A033045, A033046, A007088, A033047, A033048, A033049, A033050, A033051, A033052 are similar sequences for 2-16.

A063013 is similar in a different way.

a:= proc(n) `if`(n<2, n, irem(n, 2, 'r')+20*a(r)) end:
seq(a(n), n=0..37);  # Alois P. Heinz, Apr 04 2025

Table[FromDigits[IntegerDigits[n,2],20],{n,0,40}] (* Harvey P. Dale, Jul 21 2014 *)

baseE(x, b)= { local(d, e, f); e=0; f=1; while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) }
baseI(x, b)= { local(d, e, f); e=0; f=1; while (x>0, d=x-10*(x\10); x\=10; e+=d*f; f*=b); return(e) }
{ for (n=0, 1000, write("b063012.txt", n, " ", baseI(baseE(n, 2), 20)) ) } \\ Harry J. Smith, Aug 15 2009

def A063012(n): return int(bin(n)[2:],20) # Chai Wah Wu, Apr 04 2025

a(n) = a(n-2^floor(log_2(n))) + 20^floor(log_2(n)). a(2n) = 20*a(n); a(2n+1) = a(2n)+1 = 20*a(n)+1.
a(n) = Sum_{k>=0} A030308(n,k)*A009964(k). - Philippe Deléham, Oct 15 2011
G.f.: (1/(1 - x))*Sum_{k>=0} 20^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Edited by Charles R Greathouse IV, Aug 02 2010

A180725 Smallest power of 20 that begins with n.

Original entry on oeis.org

1, 20, 3200000, 400, 512000000000, 64000000, 703687441776640000000000000000000000000000000000000000000000, 8000, 900719925474099200000000000000000000000000000000000000000000000000000

Offset: 1

Daniel Mondot, Sep 18 2010

Cf. A009964, A180705, A180726, A180727.

A013894 a(n) = 20^(5*n + 1).

Original entry on oeis.org

20, 64000000, 204800000000000, 655360000000000000000, 2097152000000000000000000000, 6710886400000000000000000000000000, 21474836480000000000000000000000000000000, 68719476736000000000000000000000000000000000000, 219902325555200000000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (3200000).

Subsequence of A009964.

Cf. A013822, A013854, A016861.

[20^(5*n+1): n in [0..10]]; // Vincenzo Librandi, May 27 2011

NestList[3200000*# &, 20, 10] (* Paolo Xausa, Jul 13 2025 *)

a(n) = 3200000*a(n-1), a(0)=20. - Vincenzo Librandi, May 27 2011
From Elmo R. Oliveira, Jul 11 2025: (Start)
G.f.: 20/(1-3200000*x).
E.g.f.: 20*exp(3200000*x).
a(n) = A013822(n)*A013854(n) = A009964(A016861(n)). (End)

A013896 a(n) = 20^(5*n + 3).

Original entry on oeis.org

8000, 25600000000, 81920000000000000, 262144000000000000000000, 838860800000000000000000000000, 2684354560000000000000000000000000000, 8589934592000000000000000000000000000000000, 27487790694400000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..50
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (3200000).

Subsequence of A009964.

Cf. A013824, A013856, A016885.

[20^(5*n+3): n in [0..10]]; // Vincenzo Librandi, May 27 2011

20^(5Range[0,20]+3) (* or *) NestList[3200000#&,8000,20] (* Harvey P. Dale, Dec 05 2021 *)

a(n) = 3200000*a(n-1), a(0)=8000. - Vincenzo Librandi, May 27 2011
From Elmo R. Oliveira, Jul 11 2025: (Start)
G.f.: 8000/(1-3200000*x).
E.g.f.: 8000*exp(3200000*x).
a(n) = A013824(n)*A013856(n) = A009964(A016885(n)). (End)

A013766 a(n) = 20^(3*n + 1).

Original entry on oeis.org

20, 160000, 1280000000, 10240000000000, 81920000000000000, 655360000000000000000, 5242880000000000000000000, 41943040000000000000000000000, 335544320000000000000000000000000, 2684354560000000000000000000000000000, 21474836480000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (8000).

Subsequence of A009964.

Cf. A013730, A013746, A016777.

[20^(3*n+1): n in [0..10]]; // Vincenzo Librandi, Jun 27 2011

20^(3Range[0,10]+1) (* Harvey P. Dale, Jun 23 2011 *)

makelist(20^(3*n+1),n,0,20); /* Martin Ettl, Oct 21 2012 */

a(n)=20^(3*n+1) \\ Charles R Greathouse IV, Jul 10 2016

From Elmo R. Oliveira, Feb 27 2025: (Start)
G.f.: 20/(1 - 8000*x).
E.g.f.: 20*exp(8000*x).
a(n) = A013730(n)*A013746(n) = A009964(A016777(n)). (End)

A013767 a(n) = 20^(3*n + 2).

Original entry on oeis.org

400, 3200000, 25600000000, 204800000000000, 1638400000000000000, 13107200000000000000000, 104857600000000000000000000, 838860800000000000000000000000, 6710886400000000000000000000000000, 53687091200000000000000000000000000000, 429496729600000000000000000000000000000000

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (8000).

Subsequence of A009964.

Cf. A013731, A013747, A016789.

[20^(3*n+2): n in [0..10]]; // Vincenzo Librandi, Jun 27 2011

NestList[8000#&,400,10] (* Harvey P. Dale, May 11 2018 *)

a(n)=20^(3*n+2) \\ Charles R Greathouse IV, Jul 10 2016

From Elmo R. Oliveira, Feb 27 2025: (Start)
G.f.: 400/(1 - 8000*x).
E.g.f.: 400*exp(8000*x).
a(n) = A013731(n)*A013747(n) = A009964(A016789(n)). (End)

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A009984 Powers of 40.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A013725 a(n) = 20^(2*n + 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A063012 Sum of distinct powers of 20; i.e., numbers with digits in {0,1} base 20; i.e., write n in base 2 and read as if written in base 20.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A180725 Smallest power of 20 that begins with n.

Views

Author

Keywords

Crossrefs

A013894 a(n) = 20^(5*n + 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A013896 a(n) = 20^(5*n + 3).