A053539 -id:A053539 - cp's OEIS frontend

A016140 Expansion of 1/((1-3x)(1-8*x)).

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

Offset: 0

N. J. A. Sloane

In general, for expansion of 1/((1-b*x)*(1-c*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - b*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

8*a(n) gives the number of edges in the n-th-order Sierpiński carpet graph. - Eric W. Weisstein, Aug 19 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (11,-24).

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

Cf. A190543.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Table[(8^(n+1)-3^(n+1))/5, {n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)
CoefficientList[Series[1/((1-3 x)(1-8 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 24 2013 *)
LinearRecurrence[{11,-24},{1,11},30] (* Harvey P. Dale, Feb 03 2022 *)

Vec(1/((1-3*x)*(1-8*x))+O(x^30)) \\ Charles R Greathouse IV, Sep 23 2012

[lucas_number1(n,11,24) for n in range(1, 30)] # Zerinvary Lajos, Apr 27 2009

a(n) = (8^(n+1) - 3^(n+1))/5.
a(n) = 11*a(n-1) - 24*a(n-2).
a(n) = 3*a(n-1) + 8^n.
a(n) = 8*a(n-1) + 3^n.
a(n) = Sum_{i=0..n} 3^i*8^(n-i).
E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024

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)

A036294 a(n) = n*8^n.

Original entry on oeis.org

0, 8, 128, 1536, 16384, 163840, 1572864, 14680064, 134217728, 1207959552, 10737418240, 94489280512, 824633720832, 7146825580544, 61572651155456, 527765581332480, 4503599627370496, 38280596832649216, 324259173170675712, 2738188573441261568, 23058430092136939520

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018, A053539.

[n*8^n: n in [0..20]]; // Vincenzo Librandi, Aug 09 2017

Table[n 7^n, {n, 0, 30}] (* or *) LinearRecurrence[{16,-64},{0,8},31] (* Vincenzo Librandi, Aug 09 2017 *)

From Vincenzo Librandi, Aug 09 2017: (Start)
G.f.: 8*x/(8*x-1)^2.
a(n) = 16*a(n-1) - 64*a(n-2) for n > 1. (End)
From Amiram Eldar, Jul 20 2020: (Start)
Sum_{n>=1} 1/a(n) = log(8/7).
Sum_{n>=1} (-1)^(n+1)/a(n) = log(9/8). (End)
From Elmo R. Oliveira, Sep 09 2024: (Start)
E.g.f.: 8*x*exp(8*x).
a(n) = n*A001018(n) = 8*A053539(n). (End)

A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 12, 6, 1, 5, 32, 27, 8, 1, 6, 80, 108, 48, 10, 1, 7, 192, 405, 256, 75, 12, 1, 8, 448, 1458, 1280, 500, 108, 14, 1, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 10, 2304, 17496, 28672, 18750, 6480, 1372, 192, 18, 1, 11, 5120, 59049, 131072

Offset: 2

Ralf Stephan, Feb 26 2005

T(n+k,k+1) = total number of occurrences of any given letter in all possible n-length words on a k-letter alphabet. For example, with the 2 letter alphabet {0,1} there are 4 possible 2-length words: {00,01,10,11}. The letter 0 occurs 4 times altogether, as does the letter 1. T(4,3) = 4. - Ross La Haye, Jan 03 2007

Table T(n,k) = k*n^(k-1) n,k > 0 read by antidiagonals. - Boris Putievskiy, Dec 17 2012

			Triangle begins:
  1;
  2,   1;
  3,   4,    1;
  4,  12,    6,    1;
  5,  32,   27,    8,   1;
  6,  80,  108,   48,  10,   1;
  7, 192,  405,  256,  75,  12,  1;
  8, 448, 1458, 1280, 500, 108, 14, 1;

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150).
T. Mansour, Permutations containing and avoiding certain patterns, arXiv:math/9911243 [math.CO], 1999-2000.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Cf. Left-edge columns include A001787, A027471, A002697, A053464, A053469, A027473, A053539, A053540, A053541, A081127, A081128.

Table[(n - k + 1) (k - 1)^(n - k), {n, 2, 12}, {k, 2, n}] // Flatten (* Michael De Vlieger, Aug 22 2018 *)

T(n, k) = (n-k+1) * (k-1)^(n-k), k<=n.

As a linear array, the sequence is a(n) = A004736(n)*A002260(n)^(A004736(n)-1) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2))^((t*t+3*t+4)/2-n-1), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

A230539 a(n) = 3n2^(3*n-1).

Original entry on oeis.org

0, 12, 192, 2304, 24576, 245760, 2359296, 22020096, 201326592, 1811939328, 16106127360, 141733920768, 1236950581248, 10720238370816, 92358976733184, 791648371998720, 6755399441055744, 57420895248973824, 486388759756013568, 4107282860161892352

Offset: 0

Bruno Berselli, Oct 23 2013

Arithmetic derivative of 8^n: a(n) = A003415(8^n).

Sum of reciprocals of a(n), for n>0: (2/3)*log(8/7).

Bruno Berselli, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (16,-64).

Cf. A001018, A003415.
Cf. arithmetic derivative of k^n: A001787 (k=2), A027471 (k=3), A018215 (k=4), A053464 (k=5), A212700 (k=6), A027473 (k=7), this sequence, A230540 (k=9), A085708 (k=10), A081127 (k=11).
Row n=8 of A258997.

Magma
```
[3*n*2^(3*n-1): n in [0..20]];
```

A230539:=n->3*n*2^(3*n-1): seq(A230539(n), n=0..30); # Wesley Ivan Hurt, May 03 2017

Table[3 n 2^(3 n - 1), {n,0,20}]
LinearRecurrence[{16,-64},{0,12},20] (* Harvey P. Dale, Dec 25 2022 *)

a(n) = 3*n*2^(3*n-1); \\ Michel Marcus, Oct 23 2013

G.f.: 12*x/(1-8*x)^2.

a(n) = 12*A053539(n).

A317028 Triangle read by rows: T(0,0) = 1; T(n,k) = 8 * T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 8, 64, 1, 512, 16, 4096, 192, 1, 32768, 2048, 24, 262144, 20480, 384, 1, 2097152, 196608, 5120, 32, 16777216, 1835008, 61440, 640, 1, 134217728, 16777216, 688128, 10240, 40, 1073741824, 150994944, 7340032, 143360, 960, 1, 8589934592, 1342177280, 75497472, 1835008, 17920, 48

Offset: 0

Zagros Lalo, Jul 19 2018

The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013615 ((1+8*x)^n) and along skew diagonals pointing top-right in center-justified triangle given in A038279 ((8+x)^n).
The coefficients in the expansion of 1/(1-8x-x^2) are given by the sequence generated by the row sums.
The row sums are Denominators of continued fraction convergents to sqrt(17), see A041025.
If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 8.12310562561766054982... (a metallic mean), when n approaches infinity (see A176458: (4+sqrt(17))).

			Triangle begins:
1;
8;
64, 1;
512, 16;
4096, 192, 1;
32768, 2048, 24;
262144, 20480, 384, 1;
2097152, 196608, 5120, 32;
16777216, 1835008, 61440, 640, 1;
134217728, 16777216, 688128, 10240, 40;
1073741824, 150994944, 7340032, 143360, 960, 1;
8589934592, 1342177280, 75497472, 1835008, 17920, 48;
68719476736, 11811160064, 754974720, 22020096, 286720, 1344, 1;
549755813888, 103079215104, 7381975040, 251658240, 4128768, 28672, 56;
4398046511104, 893353197568, 70866960384, 2768240640, 55050240, 516096, 1792, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, Pages 70, 98

Zagros Lalo, Left-justified triangle
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (1 + 8x)^n
Zagros Lalo, Skew diagonals in center-justified triangle of coefficients in expansion of (8 + x)^n

Row sums give A041025.
Cf. A013615, A038279, A176458.
Cf. A001018 (column 0), A053539 (column 1), A081138 (column 2), A140802 (column 3), A172510 (column 4).

t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 8 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten

T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, 8*T(n-1, k)+T(n-2, k-1)));
tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n, k), ", ")); print); \\ Michel Marcus, Jul 20 2018

A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 3, 4, 1, 0, 0, 4, 12, 6, 1, 0, 0, 5, 32, 27, 8, 1, 0, 0, 6, 80, 108, 48, 10, 1, 0, 0, 7, 192, 405, 256, 75, 12, 1, 0, 0, 8, 448, 1458, 1280, 500, 108, 14, 1, 0, 0, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 0, 0, 10, 2304

Offset: 0

Franck Maminirina Ramaharo, Oct 14 2018

T(n,k) is the number of length n*k binary words of n consecutive blocks of length k, respectively, one of the blocks having exactly k letters 1, and the other having exactly one letter 0. First column follows from the next definition.
In Kauffman's language, T(n,k) is the total number of Jordan trails that are obtained by placing state markers at the crossings of the Pretzel universe P(k, k, ..., k) having n tangles, of k half-twists respectively. In other words, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) such that the final diagram is a single Jordan curve. The aforementionned binary words encode these operations by assigning each tangle a length k binary words with the adequate choice for splitting the crossings.
Columns are linear recurrence sequences with signature (2*k, -k^2).

			Square array begins:
    0, 0,   0,    0,     0,      0,      0,      0, ...
    1, 1,   1,    1,     1,      1,      1,      1, ...
    0, 2,   4,    6,     8,     10,     12,     14, ... A005843
    0, 3,  12,   27,    48,     75,    108,    147, ... A033428
    0, 4,  32,  108,   256,    500,    864,   1372, ... A033430
    0, 5,  80,  405,  1280,   3125,   6480,  12005, ... A269792
    0, 6, 192, 1458,  6144,  18750,  46656, 100842, ...
    0, 7, 448, 5103, 28672, 109375, 326592, 823543, ...
    ...
T(3,2) = 3*2^(3 - 1) = 12. The corresponding binary words are 110101, 110110, 111001, 111010, 011101, 011110, 101101, 101110, 010111, 011011, 100111, 101011.

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Alexander Stoimenow, Everywhere Equivalent 2-Component Links, Symmetry Vol. 7 (2015), 365-375.
Wikipedia, Pretzel link

Antidiagonal sums: A101495.
Column 1 is column 2 of A300453.
Column 2 is column 1 of A300184.
Cf. A104002, A320530.

T[n_, k_] = If [k > 0, n*k^(n - 1), If[k == 0 && n == 1, 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 12}]//Flatten

T(n, k) := if k > 0 then n*k^(n - 1) else if k = 0 and n = 1 then 1 else 0$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, nn))$

T(n,k) = (2*k)*T(n-1,k) - (k^2)*T(n-2,k).
G.f. for columns: x/(1 - k*x)^2.
E.g.f. for columns: x*exp(k*x).
T(n,1) = A001477(n).
T(n,2) = A001787(n).
T(n,3) = A027471(n+1).
T(n,4) = A002697(n).
T(n,5) = A053464(n).
T(n,6) = A053469(n), n > 0.
T(n,7) = A027473(n), n > 0.
T(n,8) = A053539(n).
T(n,9) = A053540(n), n > 0.
T(n,10) = A053541(n), n > 0.
T(n,11) = A081127(n).
T(n,12) = A081128(n).

cp's OEIS Frontend

A016140 Expansion of 1/((1-3*x)*(1-8*x)).

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

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A036294 a(n) = n*8^n.

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A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.

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A230539 a(n) = 3*n*2^(3*n-1).

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A317028 Triangle read by rows: T(0,0) = 1; T(n,k) = 8 * T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0.

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A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

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A016140 Expansion of 1/((1-3x)(1-8*x)).

A230539 a(n) = 3n2^(3*n-1).