A016140 -id:A016140 - cp's OEIS frontend

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

1, 2, 3, 4, 6, 9, 8, 12, 18, 27, 16, 24, 36, 54, 81, 32, 48, 72, 108, 162, 243, 64, 96, 144, 216, 324, 486, 729, 128, 192, 288, 432, 648, 972, 1458, 2187, 256, 384, 576, 864, 1296, 1944, 2916, 4374, 6561, 512, 768, 1152, 1728, 2592, 3888, 5832, 8748, 13122, 19683

Offset: 0

N. J. A. Sloane

The triangle pertaining to this sequence has the property that every row, every column and every diagonal contains a nontrivial geometric progression. More interestingly every line joining any two elements contains a nontrivial geometric progression. - Amarnath Murthy, Jan 02 2002
Kappraff states (pp. 148-149): "I shall refer to this as Nicomachus' table since an identical table of numbers appeared in the Arithmetic of Nicomachus of Gerasa (circa 150 A.D.)" The table was rediscovered during the Italian Renaissance by Leon Battista Alberti, who incorporated the numbers in dimensions of his buildings and in a system of musical proportions. Kappraff states "Therefore a room could exhibit a 4:6 or 6:9 ratio but not 4:9. This ensured that ratios of these lengths would embody musical ratios". - Gary W. Adamson, Aug 18 2003
After Nichomachus and Alberti several Renaissance authors described this table. See for instance Pierre de la Ramée in 1569 (facsimile of a page of his Arithmetic Treatise in Latin in the links section). - Olivier Gérard, Jul 04 2013
The triangle sums, see A180662 for their definitions, link Nicomachus's table with eleven different sequences, see the crossrefs. It is remarkable that these eleven sequences can be described with simple elegant formulas. The mirror of this triangle is A175840. - Johannes W. Meijer, Sep 22 2010
The diagonal sums Sum_{k} T(n - k, k) give A167762(n + 2). - Michael Somos, May 28 2012
Where d(n) is the divisor count function, then d(T(i,j)) = A003991, the rows of which sum to the tetrahedral numbers A000292(n+1). For example, the sum of the divisors of row 4 of this triangle (i = 4), gives d(16) + d(24) + d(36) + d(54) + d(81) = 5 + 8 + 9 + 8 + 5 = 35 = A000292(5). In fact, where p and q are distinct primes, the aforementioned relationship to the divisor function and tetrahedral numbers can be extended to any triangle of numbers in which the i-th row is of form {p^(i-j)*q^j, 0<=j<=i}; i >= 0 (e.g., A003593, A003595). - Raphie Frank, Nov 18 2012, corrected Dec 07 2012
Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then 2*x and 3*x are in S, and duplicates are deleted as they occur; see A232559. - Clark Kimberling, Nov 28 2013
Partial sums of rows produce Stirling numbers of the 2nd kind: A000392(n+2) = Sum_{m=1..(n^2+n)/2} a(m). - Fred Daniel Kline, Sep 22 2014
A permutation of A003586. - L. Edson Jeffery, Sep 22 2014
Form a word of length i by choosing a (possibly empty) word on alphabet {0,1} then concatenating a word of length j on alphabet {2,3,4}. T(i,j) is the number of such words. - Geoffrey Critzer, Jun 23 2016
Form of Zorach additive triangle (see A035312) where each number is sum of west and northwest numbers, with the additional condition that each number is GCD of the two numbers immediately below it. - Michel Lagneau, Dec 27 2018

			The start of the sequence as a triangular array read by rows:
   1
   2   3
   4   6   9
   8  12  18  27
  16  24  36  54  81
  32  48  72 108 162 243
  ...
The start of the sequence as a table T(n,k) n, k > 0:
    1    2    4    8   16   32 ...
    3    6   12   24   48   96 ...
    9   18   36   72  144  288 ...
   27   54  108  216  432  864 ...
   81  162  324  648 1296 2592 ...
  243  486  972 1944 3888 7776 ...
  ...
- _Boris Putievskiy_, Jan 08 2013

Jay Kappraff, Beyond Measure, World Scientific, 2002, p. 148.
Flora R. Levin, The Manual of Harmonics of Nicomachus the Pythagorean, Phanes Press, 1994, p. 114.

Reinhard Zumkeller and Matthew House, Rows n = 0..300 of triangle, flattened [Rows 0 through 120 were computed by Reinhard Zumkeller; rows 121 through 300 by Matthew House, Jul 09 2015]
Fred Daniel Kline, How do I convert this Nicomachus' Triangle to one with edges?
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Pierre de la Ramée (Petrus Ramus), P. Rami Arithmeticae (anno 1569) Liber 2, Cap. XVI "De inventione continue proportionalium" p.46 (leaf 0055) describes this integer triangle in a layout close to the current OEIS 'tabl' layout.
Marko Riedel, Proof of identity by Egorychev method.
Thomas Scheuerle, Version of this triangle from Boethius (480-524), Anicius Manlius Severinus Boethius, De institutione arithmetica, Medeltidshandskrift 1 (Mh 1), Lund University Library, early 10th century, page 4r.
Robert Sedgewick, Analysis of shellsort and related algorithms, Fourth European Symposium on Algorithms, Barcelona, September, 1996.

Cf. A001047 (row sums), A000400 (central terms), A013620, A007318.
Cf. A003586, A000079, A000244, A007283, A025197, A005010, A003946, A005051.
Triangle sums (see the comments): A001047 (Row1); A015441 (Row2); A005061 (Kn1, Kn4); A016133 (Kn2, Kn3); A016153 (Fi1, Fi2); A016140 (Ca1, Ca4); A180844 (Ca2, Ca3); A180845 (Gi1, Gi4); A180846 (Gi2, Gi3); A180847 (Ze1, Ze4); A016185 (Ze2, Ze3). - Johannes W. Meijer, Sep 22 2010, Sep 10 2011
Antidiagonal cumulative sum: A000392; square arrays cumulative sum: A160869. Antidiagonal products: 6^A000217; antidiagonal cumulative products: 6^A000292; square arrays products: 6^A005449; square array cumulative products: 6^A006002.

a036561 n k = a036561_tabf !! n !! k
a036561_row n = a036561_tabf !! n
a036561_tabf = iterate (\xs@(x:_) -> x * 2 : map (* 3) xs) [1]
-- Reinhard Zumkeller, Jun 08 2013

/* As triangle: */ [[(2^(i-j)*3^j)/3: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Oct 17 2014

A036561 := proc(n,k): 2^(n-k)*3^k end:
seq(seq(A036561(n,k),k=0..n),n=0..9);
T := proc(n,k) option remember: if k=0 then 2^n elif k>=1 then procname(n,k-1) + procname(n-1,k-1) fi: end: seq(seq(T(n,k),k=0..n),n=0..9);
# Johannes W. Meijer, Sep 22 2010, Sep 10 2011

Flatten[Table[ 2^(i-j) 3^j, {i, 0, 12}, {j, 0, i} ]] (* Flatten added by Harvey P. Dale, Jun 07 2011 *)

for(i=0,9,for(j=0,i,print1(3^j<<(i-j)", "))) \\ Charles R Greathouse IV, Dec 22 2011

{T(n, k) = if( k<0 || k>n, 0, 2^(n - k) * 3^k)} /* Michael Somos, May 28 2012 */

T(n,k) = A013620(n,k)/A007318(n,k). - Reinhard Zumkeller, May 14 2006
T(n,k) = T(n,k-1) + T(n-1,k-1) for n>=1 and 1<=k<=n with T(n,0) = 2^n for n>=0. - Johannes W. Meijer, Sep 22 2010
T(n,k) = 2^(k-1)*3^(n-1), n, k > 0 read by antidiagonals. - Boris Putievskiy, Jan 08 2013
a(n) = 2^(A004736(n)-1)*3^(A002260(n)-1), n > 0, or a(n) = 2^(j-1)*3^(i-1) n > 0, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor[(-1+sqrt(8*n-7))/2]. - Boris Putievskiy, Jan 08 2013
G.f.: 1/((1-2x)(1-3yx)). - Geoffrey Critzer, Jun 23 2016
T(n,k) = (-1)^n * Sum_{q=0..n} (-1)^q * C(k+3*q, q) * C(n+2*q, n-q). - Marko Riedel, Jul 01 2024

A016185 a(n) = 9^n - 8^n.

Original entry on oeis.org

0, 1, 17, 217, 2465, 26281, 269297, 2685817, 26269505, 253202761, 2413042577, 22791125017, 213710059745, 1992110014441, 18478745943857, 170706760005817, 1571545212141185, 14425381885981321, 132080236787517137, 1206736529597136217, 11004743954450081825

Offset: 0

N. J. A. Sloane

a(n) is also the number of n-digit numbers whose smallest decimal digit is 1. - Stefano Spezia, Nov 15 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
John Elias, Illustration of initial terms: Sierpinski Carpet
Wikipedia, Sierpinski carpet
Index entries for linear recurrences with constant coefficients, signature (17,-72).

Cf. A001018, A001019, A016140.

[9^n -8^n: n in [0..40]]; // G. C. Greubel, Nov 29 2024

LinearRecurrence[{17,-72}, {0,1}, 31] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)

def A016185(n): return pow(9,n) - pow(8,n)
print([A016185(n) for n in range(41)]) # G. C. Greubel, Nov 29 2024

G.f.: x/((1-8*x)*(1-9*x)).
E.g.f.: e^(9*x) - e^(8*x). - Mohammad K. Azarian, Jan 14 2009
a(n) = 9*a(n-1) + 8^(n-1), a(0)=0. - Vincenzo Librandi, Feb 09 2011
a(n) = 17*a(n-1) - 72*a(n-2), a(0)=0, a(1)=1. - Vincenzo Librandi, Feb 09 2011
a(n) = A001019(n) - A001018(n). - Alois P. Heinz, Aug 19 2021

A016177 a(n) = 8^n - 7^n.

Original entry on oeis.org

0, 1, 15, 169, 1695, 15961, 144495, 1273609, 11012415, 93864121, 791266575, 6612607849, 54878189535, 452866803481, 3719823438255, 30436810578889, 248242046141055, 2019169299698041, 16385984911571535, 132716292890482729, 1073129238309234975, 8664826172771491801

Offset: 0

N. J. A. Sloane

Number of ways to assign truth values to n ternary conjunctions connected by disjunctions such that the proposition is true. For example, a(2) = 15, since for the proposition '(a & b & c) v (d & e & f)' there are 15 assignments that make the proposition true. - Ori Milstein, Dec 22 2022
Equivalently, the number of length-n words over the alphabet {0,1,...,7} with at least one letter = 7. - Joerg Arndt, Jan 01 2023
a(n) is also the number of n-digit numbers whose smallest decimal digit is 2. - Stefano Spezia, Nov 15 2023

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (15,-56).

Cf. A000420, A001018, A016140.

[8^n -7^n: n in [0..40]]; // G. C. Greubel, Nov 29 2024

Table[8^n - 7^n, {n, 0, 20}] (* Harvey P. Dale, Jan 31 2011 *)

a(n)=8^n-7^n \\ Charles R Greathouse IV, Sep 28 2015

def A016177(n): return pow(8,n) - pow(7,n)
print([A016177(n) for n in range(41)]) # G. C. Greubel, Nov 29 2024

G.f.: x/((1-7x)*(1-8x)).
a(n) = numerator(f(n-1)) where f(n) = Integral_{x=0..1/4} (1-x/2)^n dx. And denominator(f(n)) = 4*(n+1)*8^n. - Al Hakanson (hawkuu(AT)excite.com), Feb 22 2004 [corrected by Michel Marcus, Dec 23 2022]
a(n) = 15*a(n-1) - 56*a(n-2), n > 1. - Philippe Deléham, Jan 01 2009
E.g.f.: e^(8*x) - e^(7*x). - Mohammad K. Azarian, Jan 14 2009
a(n) = 8*a(n-1) + 7^(n-1), a(0)=0. - Vincenzo Librandi, Feb 09 2011

A271939 Number of edges in the n-th order Sierpinski carpet graph.

Original entry on oeis.org

8, 88, 776, 6424, 52040, 418264, 3351944, 26833048, 214716872, 1717892440, 13743611912, 109950312472, 879606751304, 7036866765016, 56294972383880, 450359893862296, 3602879495272136, 28823036995298392, 230584299061751048, 1844674401792100120

Offset: 1

Emeric Deutsch, Apr 17 2016

Also the number of maximal and maximum cliques in the n-Sierpinski carpet graph. - Eric W. Weisstein, Dec 01 2017

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

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, Edge Count
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Cf. A016140.

Cf. A001018 (number of vertices in the n-Sierpinski carpet graph).

Maple
```
seq((1/5)*(8*(8^n-3^n)), n = 1 .. 20);
```

Table[8 (8^n - 3^n)/5, {n, 20}] (* Eric W. Weisstein, Jun 17 2017 *)
LinearRecurrence[{11, -24}, {8, 88}, 20] (* Eric W. Weisstein, Jun 17 2017 *)
CoefficientList[Series[8/(1 - 11 x + 24 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 17 2017 *)

x='x+O('x^99); Vec(8/((1-3*x)*(1-8*x))) \\ Altug Alkan, Apr 17 2016

a(n) = 8 * (8^n - 3^n)/5.
a(n) = 8 * A016140(n).
G.f.: 8*x / ( (8*x-1)*(3*x-1) ). - R. J. Mathar, Apr 17 2016
a(n) = 8*a(n-1) + 8*3^(n-1). - Allan Bickle, Nov 27 2022

A016186 Expansion of 1/((1-8x)(1-10*x)).

Original entry on oeis.org

1, 18, 244, 2952, 33616, 368928, 3951424, 41611392, 432891136, 4463129088, 45705032704, 465640261632, 4725122093056, 47800976744448, 482407813955584, 4859262511644672, 48874100093157376, 490992800745259008, 4927942405962072064, 49423539247696576512, 495388313981572612096, 4963106511852580896768

Offset: 0

N. J. A. Sloane

a(n) is the number of strings of n+1 decimal digits having an odd number of 0's. For 2 digits these are for example the 18 strings 01, 02, 03, ..., 09, 10, 20, 30, ..., 90. - Geoffrey Critzer, Jan 24 2011

G. C. Greubel, Table of n, a(n) for n = 0..990
Index entries for linear recurrences with constant coefficients, signature (18,-80).

Cf. A016140, A060531, A081035, A081203.

[2^n*(5^(n+1)-4^(n+1)): n in [0..40]]; // G. C. Greubel, Nov 14 2024

Rest@With[{m=30}, CoefficientList[Series[Exp[9 x] Sinh[x], {x,0,m}], x]*Range[0, m]!]
Table[2^n*(5^(n+1)-4^(n+1)), {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18,-80},{1,18},30] (* Harvey P. Dale, Aug 26 2019 *)

Vec(1/((1-8*x)*(1-10*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 24 2012

A016186=BinaryRecurrenceSequence(18,-80,1,18)
print([A016186(n) for n in range(41)]) # G. C. Greubel, Nov 14 2024

From R. J. Mathar, Sep 18 2008: (Start)
a(n) = 5*10^n - 4*8^n = A081203(n+1).
Binomial transform of A081035. (End)
From Geoffrey Critzer, Jan 24 2011: (Start)
a(n) = 8*a(n-1) + 10^(n-1).
E.g.f.: exp(9*x)*sinh(x) (with offset 1). (End)
A060531(n) = a(n) - 9*a(n-1). - R. J. Mathar_, Jan 27 2011
From Vincenzo Librandi, Feb 09 2011: (Start)
a(n) = 10*a(n-1) + 8^n, a(0)=1.
a(n) = 18*a(n-1) - 80*a(n-2), a(0)=1, a(1)=18. (End)
E.g.f.: exp(9*x)*( cosh(x) + 9*sinh(x) ). - G. C. Greubel, Nov 14 2024

More terms added by G. C. Greubel, Nov 14 2024

A180845 a(n) = (16^n-3^n)/13.

Original entry on oeis.org

0, 1, 19, 313, 5035, 80641, 1290499, 20648713, 330381595, 5286112081, 84577812979, 1353245066713, 21651921244555, 346430740444321, 5542891848703459, 88686269584038313, 1418980313358961915

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) appear in several triangle sums of Nicomachus' table A036561, i.e Gi1(4*n), Gi1(4*n+1)/2, Gi1(4*n+2)/4, Gi1(4*n+3)/8 and Gi4(n). See A180662 for information about these giraffe and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (19, -48).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

Table[(16^n-3^n)/13,{n,0,30}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011*)

a(n) = (16^n-3^n)/13

G.f.: x/((16*x-1)*(3*x-1))

A180844 a(n) = (27^n - 2^n)/25.

Original entry on oeis.org

0, 1, 29, 787, 21257, 573955, 15496817, 418414123, 11297181449, 305023899379, 8235645283745, 222362422662139, 6003785411879801, 162102206120758723, 4376759565260493713, 118172508262033346635, 3190657723074900391913

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) appear in several triangle sums of Nicomachus's table A036561, i.e., Ca2(3*n), Ca2(3*n+1)/3, Ca2(3*n+2)/9 and Ca3(n). See A180662 for information about these camel sums and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (29, -54).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

(#[[1]]-#[[2]])/25&/@Partition[Riffle[27^Range[0,20],2^Range[0,20]],2]  (* Harvey P. Dale, Jan 22 2011 *)

a(n) = (27^n - 2^n)/25 \\ Iain Fox, Dec 12 2017

first(n) = Vec(x/((27*x-1)*(2*x-1)) + O(x^n), -n) \\ Iain Fox, Dec 12 2017

a(n) = (27^n - 2^n)/25.

G.f.: x/((27*x-1)*(2*x-1)).

A180846 a(n) = (81^n - 2^n)/79.

Original entry on oeis.org

0, 1, 83, 6727, 544895, 44136511, 3575057423, 289579651327, 23455951757615, 1899932092367071, 153894499481733263, 12465454458020395327, 1009701811099652023535, 81785846699071813910431, 6624653582624816926753103

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) appear in several triangle sums of Nicomachus's table A036561, i.e., Gi2(4*n), Gi2(4*n+1)/2, Gi2(4*n+2)/4, Gi2(4*n+3)/8 and Gi3(n). See A180662 for information about these giraffe and other chess sums.

Nathaniel Johnston, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (83,-162).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

[(81^n-2^n)/79: n in [0..50]]; // Vincenzo Librandi, Apr 15 2011

Table[(81^n-2^n)/79,{n,0,15}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2011 *)

a(n) = (81^n - 2^n)/79.

G.f.: x/((81*x-1)*(2*x-1)).

A180847 a(n) = (27^n-4^n)/23.

Original entry on oeis.org

0, 1, 31, 853, 23095, 623821, 16844191, 454797253, 12279542215, 331547705341, 8951788306351, 241698285320053, 6525853707835735, 176198050128342061, 4757347353532344511, 128448378545641737253, 3468106220733400647655

Offset: 0

Johannes W. Meijer, Sep 21 2010

For n>0, a(n) appears in several triangle sums of Nicomachus' table A036561, i.e. Ze1(2*n), Ze1(2*n+1)/2; Ze4(3*n), Ze4(3*n+1)/3 and Ze4(3*n+2)/9. See A180662 for information about these zebra and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (31,-108).

Cf. A016153, A016140, A180844, A180845, A180846, A180847, A016185.

Table[(27^n-4^n)/23,{n,0,20}] (* or *) LinearRecurrence[{31,-108},{0,1},20]  (* Harvey P. Dale, Sep 01 2011 *)

a(n)=(27^n-4^n)/23 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = (27^n-4^n)/23.
G.f.: x/((27*x-1)*(4*x-1)).
a(0)=0, a(1)=1, a(n) = 31*a(n-1)-108*a(n-2). - Harvey P. Dale, Sep 01 2011

A016187 Expansion of 1/((1-8x)(1-11*x)).

Original entry on oeis.org

1, 19, 273, 3515, 42761, 503139, 5796673, 65860555, 741243321, 8287894259, 92240578673, 1023236299995, 11324318776681, 125117262357379, 1380687932442273, 15222751628953835, 167731742895202841, 1847300971660916499, 20338325086779563473, 223865691142651054075, 2463675524073768441801, 27109654136848307635619

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..950
Index entries for linear recurrences with constant coefficients, signature (19,-88).

Cf. A016140.

[(11^(n+1)-8^(n+1))/3: n in [0..40]]; // G. C. Greubel, Nov 14 2024

Table[(11^(n+1)-8^(n+1))/3, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)
LinearRecurrence[{19,-88}, {1,19}, 40] (* G. C. Greubel, Nov 14 2024 *)

PARI
```
for(n=1,10,print1((11^n-8^n)/3,","))
```

MM(n, N) = local(M); M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M
for(i=1,10,print1((MM(3,9)^i)[1,2],","))

A016187=BinaryRecurrenceSequence(19,-88,1,19)
print([A016187(n) for n in range(41)]) # G. C. Greubel, Nov 14 2024

a(n) = (11^(n+1) - 8^(n+1))/3. - Lambert Klasen (lambert.klasen(AT)gmx.net), Feb 05 2005
a(n) = 11*a(n-1) + 8^n, a(0)=1. - Vincenzo Librandi, Feb 09 2011
a(n) = 19*a(n-1) - 88*a(n-2), a(0)=1, a(1)=19. - Vincenzo Librandi, Feb 09 2011
E.g.f.: (1/3)*(11*exp(11*x) - 8*exp(8*x)). - G. C. Greubel, Nov 14 2024

More terms added by G. C. Greubel, Nov 14 2024

cp's OEIS Frontend

A036561 Nicomachus triangle read by rows, T(n, k) = 2^(n - k)*3^k, for 0 <= k <= n.

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A016185 a(n) = 9^n - 8^n.

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A016177 a(n) = 8^n - 7^n.

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A271939 Number of edges in the n-th order Sierpinski carpet graph.

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A016186 Expansion of 1/((1-8*x)*(1-10*x)).

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A180845 a(n) = (16^n-3^n)/13.

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A016186 Expansion of 1/((1-8x)(1-10*x)).

A016187 Expansion of 1/((1-8x)(1-11*x)).