A016187 -id:A016187 - 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

A139746 a(n) = 11^n - 8^n.

Original entry on oeis.org

0, 3, 57, 819, 10545, 128283, 1509417, 17390019, 197581665, 2223729963, 24863682777, 276721736019, 3069708899985, 33972956330043, 375351787072137, 4142063797326819, 45668254886861505, 503195228685608523, 5541902914982749497, 61014975260338690419, 671597073427953162225

Offset: 0

N. J. A. Sloane, May 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (19,-88).

Cf. A016187.

[11^n-8^n: n in [0..30]]; // Vincenzo Librandi, Jun 02 2011

Table[11^n-8^n,{n,0,30}] (* or *) LinearRecurrence[{19,-88},{0,3},30] (* Harvey P. Dale, Apr 13 2019 *)

a(n) = 19*a(n-1) - 88*a(n-2). - Vincenzo Librandi, Jun 02 2011
From Elmo R. Oliveira, Apr 01 2025: (Start)
G.f.: 3*x/((1-8*x)*(1-11*x)).
E.g.f.: 2*exp(19*x/2)*sinh(3*x/2).
a(n) = 3*A016187(n-1) for n >= 1. (End)

A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 3, 0, 1, 5, 9, 5, 0, 1, 7, 21, 27, 11, 0, 1, 9, 39, 85, 81, 21, 0, 1, 11, 63, 203, 341, 243, 43, 0, 1, 13, 93, 405, 1031, 1365, 729, 85, 0, 1, 15, 129, 715, 2511, 5187, 5461, 2187, 171, 0, 1, 17, 171, 1157, 5261, 15309, 25999, 21845, 6561, 341, 0, 1

Offset: 0

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 09 2005

Consider a 3 X 3 matrix M =
  [n, 1, 1]
  [1, n, 1]
  [1, 1, n].
The n-th row of the array contains the values of the nondiagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = nondiagonal entry + (n-1)^k.)
Table:
  n: row sequence G.f. cross references.
  0: (2^n-(-1)^n)/3 1/((1+1x)(1-2x)) A001045 (Jacobsthal sequence)
  1: (3^n-0^n)/3 1/(1-3x) A000244
  2: (4^n-1^n)/3 1/((1-1x)(1-4x)) A002450
  3: (5^n-2^n)/3 1/((1-2x)(1-5x)) A016127
  4: (6^n-3^n)/3 1/((1-3x)(1-6x)) A016137
  5: (7^n-4^n)/3 1/((1-4x)(1-7x)) A016150
  6: (8^n-5^n)/3 1/((1-5x)(1-8x)) A016162
  7: (9^n-6^n)/3 1/((1-6x)(1-9x)) A016172
  8: (10^n-7^n)/3 1/((1-7x)(1-10x)) A016181
  9: (11^n-8^n)/3 1/((1-8x)(1-11x)) A016187
  10:(12^n-9^n)/3 1/((1-9x)(1-12x)) A016191
If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+2.
Columns follow polynomials with certain binomial coefficients:
  column: polynomial
  0: 0
  1: 1
  2: 2*n + 1
  3: 3*n^2+ 3*n + 3
  4: 4*n^3+ 6*n^2+ 12*n + 5
  5: 5*n^4+10*n^3+ 30*n^2+ 25*n + 11
  6: 6*n^5+15*n^4+ 60*n^3+ 75*n^2+ 66*n + 21
  7: 7*n^6+21*n^5+105*n^4+ 175*n^3+ 231*n^2+ 147*n + 43
  8: 8*n^7+28*n^6+168*n^5+ 350*n^4+ 616*n^3+ 588*n^2+344*n+ 85
  etc.
Coefficients are generated by the array T(n,k)=(2^(n-k-1)-(-1)^(n-k-1))/3*(binomial(k+(n-k-1),n-k-1)) read by antidiagonals.

			Array begins:
  0, 1, 1,  3,   5,   11, ...
  0, 1, 3,  9,  27,   81, ...
  0, 1, 5, 21,  85,  341, ...
  0, 1, 7, 39, 203, 1031, ...
  0, 1, 9, 63, 405, 2511, ...
  ...

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(k=0,10, for(i=0,10,print1((MM(3,k)^i)[1,2],","));print())

cp's OEIS Frontend

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

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A139746 a(n) = 11^n - 8^n.

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A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A139746 a(n) = 11^n - 8^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

Views

Author

Keywords

Comments

Examples

Programs

PARI

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