A001018 -id:A001018 - 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

A049306 Numbers k such that k is a substring of 8^k.

Original entry on oeis.org

4, 6, 7, 10, 13, 17, 18, 28, 31, 33, 36, 38, 42, 44, 47, 48, 49, 52, 54, 56, 58, 60, 63, 64, 67, 68, 69, 76, 77, 79, 81, 82, 83, 85, 86, 89, 90, 91, 94, 97, 112, 115, 124, 130, 135, 165, 173, 176, 178, 189, 193, 195, 206, 208, 215, 221, 225, 249, 251, 252, 253, 256

Offset: 1

David W. Wilson

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001018, A032740, A049301, A049302, A049303, A049304, A049305, A049307.

Select[Range[300],SequenceCount[IntegerDigits[8^#],IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2018 *)

def ok(n): return str(n) in str(8**n)
print(list(filter(ok, range(257)))) # Michael S. Branicky, Aug 13 2021

A128967 a(n) = (n^3-n)*8^n.

Original entry on oeis.org

0, 384, 12288, 245760, 3932160, 55050240, 704643072, 8455716864, 96636764160, 1063004405760, 11338713661440, 117922622078976, 1200666697531392, 12006666975313920, 118219490218475520, 1148417904979476480, 11024811887802974208, 104735712934128254976

Offset: 1

Mohammad K. Azarian, Apr 28 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (32,-384,2048,-4096).

Cf. A001018, A007531, A036289, A128796, A140802.

Cf. A128960, A128961, A128962, A128963, A128964, A128965, A128969.

[(n^3 - n)*8^n: n in [1..25]]; // Vincenzo Librandi, Feb 11 2013

LinearRecurrence[{32, -384, 2048, -4096}, {0, 384, 12288, 245760}, 30] (* Vincenzo Librandi, Feb 11 2013 *)

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: 384x^2/(1-8x)^4.
a(n) = 384*A140802(n-2). (End)
a(n) = 32*a(n-1) - 384*a(n-2) + 2048*a(n-3) - 4096*a(n-4). - Vincenzo Librandi, Feb 11 2013
From Amiram Eldar, Oct 02 2022: (Start)
a(n) = A007531(n+1)*A001018(n).
Sum_{n>=2} 1/a(n) = (49/16)*log(8/7) - 13/32.
Sum_{n>=2} (-1)^n/a(n) = (81/16)*log(9/8) - 19/32. (End)

Corrected the offset. - Mohammad K. Azarian, Nov 20 2008

A183819 T(n,k)=1/81 the number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock being a reflection across the shared element pair of any horizontal or vertical neighbor.

Original entry on oeis.org

1, 8, 8, 64, 162, 64, 512, 3256, 3256, 512, 4096, 65640, 163208, 65640, 4096, 32768, 1321860, 8231592, 8231592, 1321860, 32768, 262144, 26630220, 414246824, 1042063656, 414246824, 26630220, 262144, 2097152, 536415300, 20864057480

Offset: 1

R. H. Hardin Jan 07 2011

Table starts
.........1.............8...............64................512
.........8...........162.............3256..............65640
........64..........3256...........163208............8231592
.......512.........65640..........8231592.........1042063656
......4096.......1321860........414246824.......131467169720
.....32768......26630220......20864057480.....16607782066704
....262144.....536415300....1050517114472...2096961683069192
...2097152...10805642100...52900274426120.264820293988996048
..16777216..217666494900.2663752907479720
.134217728.4384657426500

			Some solutions with upper left block zero for 4X3
..0..0..0....0..0..1....0..0..1....0..0..1....0..0..0....0..0..1....0..0..1
..0..0..2....0..0..0....0..0..2....0..0..1....0..0..1....0..0..0....0..0..1
..1..0..1....0..2..1....1..2..0....1..1..0....2..1..1....2..1..0....1..1..2
..2..1..1....1..2..2....1..1..0....0..1..1....0..2..0....0..2..2....2..0..2

R. H. Hardin, Table of n, a(n) for n = 1..70

Column 1 is A001018(n-1)

A212702 Main transitions in systems of n particles with spin 7/2.

Original entry on oeis.org

7, 112, 1344, 14336, 143360, 1376256, 12845056, 117440512, 1056964608, 9395240960, 82678120448, 721554505728, 6253472382976, 53876069761024, 461794883665920, 3940649673949184, 33495522228568064, 283726776524341248, 2395915001761103872, 20176126330619822080

Offset: 1

Stanislav Sykora, May 25 2012

Please, refer to the general explanation in A212697.

This sequence is for base b=8 (see formula), corresponding to spin S=(b-1)/2=7/2.

Stanislav Sykora, Table of n, a(n) for n = 1..100
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001787, A212697, A212698, A212699, A212700, A212701, A212703, A212704 (b = 2, 3, 4, 5, 6, 7, 9, 10).

Cf. A001018, A008589, A053539.

LinearRecurrence[{16,-64},{7,112},30] (* Harvey P. Dale, Feb 11 2016 *)

mtrans(n, b) = n*(b-1)*b^(n-1);
for (n=1, 100, write("b212702.txt", n, " ", mtrans(n, 8)))

Vec(7*x/(8*x-1)^2 + O(x^100)) \\ Colin Barker, Jun 16 2015

a(n) = n*(b-1)*b^(n-1). For this sequence, set b=8.
From Colin Barker, Jun 16 2015: (Start)
a(n) = 16*a(n-1) - 64*a(n-2) for n > 2.
G.f.: 7*x/(8*x-1)^2. (End)
From Elmo R. Oliveira, May 14 2025: (Start)
E.g.f.: 7*x*exp(8*x).
a(n) = 7*A053539(n) = A008589(n)*A001018(n-1). (End)

A233168 T(n,k)=Number of nXk 0..7 arrays with no element x(i,j) adjacent to itself or value 7-x(i,j) horizontally, diagonally, antidiagonally or vertically, top left element zero, and 1 appearing before 2 3 4 5 and 6, 2 appearing before 3 4 and 5, and 3 appearing before 4 in row major order (unlabeled 8-colorings with no clashing color pairs).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 11, 8, 8, 11, 48, 64, 96, 64, 48, 236, 512, 1280, 1280, 512, 236, 1248, 4096, 18432, 28672, 18432, 4096, 1248, 6896, 32768, 278528, 720896, 720896, 278528, 32768, 6896, 39168, 262144, 4325376, 19922944, 31457280, 19922944, 4325376

Offset: 1

R. H. Hardin, Dec 05 2013

Table starts
......1........1...........3.............11................48
......1........1...........8.............64...............512
......3........8..........96...........1280.............18432
.....11.......64........1280..........28672............720896
.....48......512.......18432.........720896..........31457280
....236.....4096......278528.......19922944........1543503872
...1248....32768.....4325376......587202560.......83751862272
...6896...262144....68157440....17985175552.....4879082848256
..39168..2097152..1082130432...562640715776...296868139499520
.226496.16777216.17246978048.17798344474624.18506979718725632

			Some solutions for n=4 k=4
..0..1..2..3....0..1..7..1....0..1..7..6....0..1..7..2....0..1..0..6
..2..4..0..1....2..3..5..3....2..3..5..4....3..2..3..1....2..3..2..4
..7..6..5..3....7..6..0..6....6..0..1..0....7..1..7..5....0..1..0..6
..2..4..0..6....3..2..3..5....4..2..3..5....4..5..4..6....3..5..3..2

R. H. Hardin, Table of n, a(n) for n = 1..544

Column 2 is A001018(n-2)

Empirical for column k:
k=1: a(n) = 12*a(n-1) -44*a(n-2) +48*a(n-3) for n>4
k=2: a(n) = 8*a(n-1) for n>2
k=3: a(n) = 24*a(n-1) -128*a(n-2) for n>3
k=4: a(n) = 48*a(n-1) -512*a(n-2) for n>3
k=5: a(n) = 96*a(n-1) -2048*a(n-2) for n>3
k=6: a(n) = 192*a(n-1) -8192*a(n-2) for n>3
k=7: a(n) = 384*a(n-1) -32768*a(n-2) for n>3

A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.

Original entry on oeis.org

0, 1, 0, 3, 4, 1, 0, 9, 24, 22, 8, 1, 0, 27, 108, 171, 136, 57, 12, 1, 0, 81, 432, 972, 1200, 886, 400, 108, 16, 1, 0, 243, 1620, 4725, 7920, 8430, 5944, 2810, 880, 175, 20, 1, 0, 729, 5832, 20898, 44280, 61695, 59472, 40636, 19824, 6855, 1640, 258, 24, 1

Offset: 0

Franck Maminirina Ramaharo, Feb 26 2018

T(n,k) is the number of state diagrams having k components of n connected summed trefoil knots.

Row sums gives A001018.

			The triangle T(n, k) begins:
n\k 0     1      2      3       4       5       6      7        8       9
0:  0     1
1:  0     3      4      1
2:  0     9     24     22       8       1
3:  0    27    108    171     136      57      12       1
4:  0    81    432    972    1200     886     400     108      16       1

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.

Michael De Vlieger, Table of n, a(n) for n = 0..10301 (rows 0 <= n <= 100, flattened.)
Ryo Hanaki, Pseudo diagrams of knots, links and spatial graphs, Osaka Journal of Mathematics, Vol. 47 (2010), 863-883.
Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Carolina Medina, Jorge Ramírez-Alfonsín and Gelasio Salazar, On the number of unknot diagrams, arXiv:1710.06470 [math.CO], 2017.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Franck Ramaharo, A bracket polynomial for 2-tangle shadows, arXiv:2002.06672 [math.CO], 2020.

Row 2: row 5 of A158454.
Row 3: row 2 of A220665.
Row 4: row 5 of A219234.

row[n_] := CoefficientList[x*(x^2 + 4*x + 3)^n, x]; Array[row, 7, 0] // Flatten (* Jean-François Alcover, Mar 16 2018 *)

g(x, y) := taylor(x/(1 - y*(x^2 + 4*x + 3)), y, 0, 10)$
a : makelist(ratcoef(g(x, y), y, n), n, 0, 10)$
T : []$
for i:1 thru 11 do
  T : append(T, makelist(ratcoef(a[i], x, n), n, 0, 2*i - 1))$
T;

T(n, k) = polcoeff(x*(x^2 + 4*x + 3)^n, k);
tabf(nn) = for (n=0, nn, for (k=0, 2*n+1, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 03 2018

T(n,k) = coefficients of x*(x^2 + 4*x + 3)^n.
T(n,k) = T(n-1,k-2) + 4*T(n-1,k-1) + 3*T(n-1,k), with T(n,0) = 0, T(n,1) = 3^n and  T(n,2) =  4*n*3^(n-1).
T(n,n+k+1) = A152842(2*n,n+k) and T(n,n-k) = A152842(2*n,n+k+1), for n >= k >= 0.
T(n,1) =  A000244(n).
T(n,2) =  A120908(n).
T(n,n+1) = A069835(n).
T(n,2*n-1) = A139272(n).
T(n,2*n) = A008586(n).
T(n,2*n-2) = A140138(4*n) = A185872(2n,2) for n >= 1.
G.f.: x/(1 - y*(x^2 + 4*x + 3)).

Typo in row 6 corrected by Jean-François Alcover, Mar 16 2018

A067412 Fourth column of triangle A067410.

Original entry on oeis.org

1, 5, 40, 320, 2560, 20480, 163840, 1310720, 10485760, 83886080, 671088640, 5368709120, 42949672960, 343597383680, 2748779069440, 21990232555520, 175921860444160, 1407374883553280, 11258999068426240

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column gives [1,6,60,600,6000,60000,...].

a(n+1) = A157176(A016957(n)). [From Reinhard Zumkeller, Feb 24 2009]

Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A067411 (third column), A067413 (sixth column), A001018 (powers of 8).

Join[{1},NestList[8#&,5,20]] (* or *) CoefficientList[Series[ (1-3x)/ (1-8x),{x,0,20}],x] (* Harvey P. Dale, May 14 2011 *)

a(n)= A067410(n+3, 3). a(n)= 5*8^(n-1), n>=1, a(0)=1.
G.f.: (1-3*x)/(1-8*x).
E.g.f.: (5*exp(8*x)+3)/8 = exp(4*x)*(cosh(4*x)+sinh(4*x)/4) - Paul Barry, Nov 20 2003

A105317 Powers of Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 13, 16, 21, 25, 27, 32, 34, 55, 64, 81, 89, 125, 128, 144, 169, 233, 243, 256, 377, 441, 512, 610, 625, 729, 987, 1024, 1156, 1597, 2048, 2187, 2197, 2584, 3025, 3125, 4096, 4181, 6561, 6765, 7921, 8192, 9261, 10946, 15625, 16384, 17711

Offset: 1

Reinhard Zumkeller, Apr 25 2005

nonn

The subset of nontrivial Fibonacci powers [numbers A000045(k)^n which are not in A000045] starts 4, 9, 16, 25, 27, 32, 64, 81, 125, 128, 169, 243, 256, 441, 512, 625, 729, 1024, 1156... - R. J. Mathar, Jan 26 2015. These are the initial terms of A254719. - Reinhard Zumkeller, Feb 06 2015

			2197 = 13^3 = A000045(7)^3, therefore 2197 is a term.

R. Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Fibonacci Number

Subsequences: A056570-A056574, A007598, A000045, A000079, A000244, A000351, A001018, A001022 and A009965.

Cf. A103323, A254719.

import Data.Set (singleton, deleteFindMin, insert)
a105317 n = a105317_list !! (n-1)
a105317_list = 0 : 1 : h 1 (drop 4 a000045_list) (singleton (2, 2)) where
  h y xs'@(x:xs) s
    | x < ff    = h y xs (insert (x, x) s)
    | ff == y   = h y xs' s'
    | otherwise = ff : h ff xs' (insert (f * ff, f) s')
    where ((ff, f), s') = deleteFindMin s
-- Reinhard Zumkeller, Feb 06 2015

N:= 10^6: # to get all terms <= N
select(`<=`,{0,1,seq(seq(combinat:-fibonacci(i)^j, i = 3 ..floor(log[phi](sqrt(5)*N^(1/j)+1))),j=1..ilog2(N))},N);
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(%,list)); # Robert Israel, Jan 26 2015

lim = 10^5; t = Table[f = Fibonacci[n]; f^Range[Floor[Log[lim]/Log[f]]], {n, 3, Ceiling[Log[GoldenRatio, lim] + 1]}]; Union[{0, 1}, Flatten[t]] (* T. D. Noe, Sep 27 2011 *)

list(lim)=my(v=List([0]),k=1,f,t); while(k<=lim, listput(v,k); k*=2); k=3; while(k<=lim, listput(v,k); k*=3); k=5; while(k<=lim, listput(v,k); k*=5); k=6; while((f=fibonacci(k++))<=lim, t=1; while((t*=f)<=lim, listput(v,t))); Set(v) \\ Charles R Greathouse IV, Oct 03 2016

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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A049306 Numbers k such that k is a substring of 8^k.

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A128967 a(n) = (n^3-n)*8^n.

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A183819 T(n,k)=1/81 the number of (n+1)X(k+1) 0..2 arrays with no 2X2 subblock being a reflection across the shared element pair of any horizontal or vertical neighbor.

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A212702 Main transitions in systems of n particles with spin 7/2.

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A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2*k+1) = A152842(2*n,2*(n-k)) and T(n,2*k) = A152842(2*n,2*(n-k)+1) for n >= k > 0.

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A299989 Triangle read by rows: T(n,0) = 0 for n >= 0; T(n,2k+1) = A152842(2n,2(n-k)) and T(n,2k) = A152842(2n,2(n-k)+1) for n >= k > 0.