A167666 -id:A167666 - cp's OEIS frontend

A167667 Expansion of (1-x+4x^2)/(1-2x)^2.

1, 3, 12, 36, 96, 240, 576, 1344, 3072, 6912, 15360, 33792, 73728, 159744, 344064, 737280, 1572864, 3342336, 7077888, 14942208, 31457280, 66060288, 138412032, 289406976, 603979776, 1258291200, 2617245696, 5435817984, 11274289152, 23353884672, 48318382080

Offset: 0

Philippe Deléham, Nov 08 2009

Also the number of maximal and maximum cliques in the n-cube-connected cycles graph for n > 3. - Eric W. Weisstein, Dec 01 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..3000
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018
Eric Weisstein's World of Mathematics, Cube-Connected Cycle Graph
Eric Weisstein's World of Mathematics, Maximal Clique
Eric Weisstein's World of Mathematics, Maximum Clique
Index entries for linear recurrences with constant coefficients, signature (4, -4).

Cf. A167666.

[1] cat [3*n*2^(n-1): n in [1..30]]; // Vincenzo Librandi, Mar 21 2015

A167667:=n->3*n*2^(n-1): (1,seq(A167667(n), n=1..30)); # Wesley Ivan Hurt, Mar 20 2015

CoefficientList[Series[(1 - x + 4*x^2)/(1 - 2*x)^2, {x, 0, 30}], x] (* Wesley Ivan Hurt, Mar 20 2015 *)
Join[{1}, LinearRecurrence[{4, -4}, {3, 12}, 20]] (* Eric W. Weisstein, Dec 01 2017 *)
Join[{1}, Table[3 2^(n - 1) n, {n, 20}]] (* Eric W. Weisstein, Dec 01 2017 *)
CoefficientList[Series[(1 - x + 4 x^2)/(-1 + 2 x)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Dec 01 2017 *)

Vec((1-x+4*x^2)/(1-2*x)^2 + O(x^50)) \\ Michel Marcus, Mar 21 2015

a(n) = if(n==0, 1, 3*n*2^(n-1)); \\ Altug Alkan, May 16 2018

a(0)=1, a(n) = 3*n*2^(n-1) for n>0.
a(0)=1, a(1)=3, a(2)=12, a(n) = 4*a(n-1)-4*a(n-2) for n>2.
a(n) = Sum_{k=0..n} A167666(n,k) * 2^k.
G.f.: 1 + 3*x*G(0)/2, where G(k)= 1 + 1/(1 - x/(x + (k+1)/(2*k+4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
a(0)=1, a(n) = Sum_{i=0..n} binomial(n,i) * (2n-i). - Wesley Ivan Hurt, Mar 20 2015

A130779 a(0)=a(1)=1, a(2)=2, a(n)=0 for n >= 3.

Original entry on oeis.org

1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Paul Curtz, Jul 14 2007

Inverse binomial transform of A002522. - R. J. Mathar, Jun 13 2008
Multiplicative with a(2)=2, a(2^e)=0 if e>1, a(p^e)=0 for odd prime p if e>=1. Dirichlet g.f. 1+2^(1-s). - R. J. Mathar, Jun 28 2011
a(n-1) is the determinant of the symmetric n X n matrix M(i,j) = rad(gcd(i,j)) for 1 <= i, j <= n, where rad(n) is the largest squarefree number dividing n (A007947). - Amiram Eldar, Jul 19 2019

J. Sándor and B. Crstici, Handbook of Number Theory II, Kluwer, 2004, p. 265.

Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A002522, A007947, A130706, A167666.

PadRight[{1,1,2},120,0] (* Harvey P. Dale, May 02 2015 *)
LinearRecurrence[{1},{1,1,2,0},105] (* Ray Chandler, Jul 15 2015 *)

a(n)=if(n<3,max(n,1),0) \\ Charles R Greathouse IV, Dec 21 2011

G.f.: 1+x+2x^2.
a(n) = A167666(n,0). - Philippe Deléham, Feb 18 2012
a(n) = n! mod 3. - Charles Kusniec, Jan 25 2020

A167682 Expansion of (1 - 2x + 5x^2) / (1 - 3*x)^2.

Original entry on oeis.org

1, 4, 20, 84, 324, 1188, 4212, 14580, 49572, 166212, 551124, 1810836, 5904900, 19131876, 61647156, 197696052, 631351908, 2008846980, 6370914708, 20145865428, 63536960196, 199908972324, 627621192180, 1966546402164, 6150687683364, 19205208480708

Offset: 0

Philippe Deléham, Nov 09 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-9).

CoefficientList[Series[(1-2x+5*x^2)/(1-3x)^2,{x,0,40}],x] (* or *) Join[{1},LinearRecurrence[{6,-9},{4,20},40]] (* Harvey P. Dale, Oct 20 2011 *)

Vec((1-2*x+5*x^2) / (1-3*x)^2 + O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(0)=1, a(1)=4, a(2)=20, a(n) = 6*a(n-1) - 9*a(n-2) for n>2.
a(n) = 4*A081038(n-1) for n>0.
a(n) = Sum_{k=0..n} A167666(n,k)*3^k.
a(n) = 3^(n - 2)*(8*n + 4) for n>0. - Colin Barker, Jan 21 2017

Corrected and extended by Harvey P. Dale, Oct 20 2011

PARI code corrected by Colin Barker, Jan 21 2017

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A167667 Expansion of (1-x+4x^2)/(1-2x)^2.

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A167667 Expansion of (1-x+4*x^2)/(1-2*x)^2.

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A130779 a(0)=a(1)=1, a(2)=2, a(n)=0 for n >= 3.

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A167682 Expansion of (1 - 2*x + 5*x^2) / (1 - 3*x)^2.

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A167667 Expansion of (1-x+4x^2)/(1-2x)^2.

A167682 Expansion of (1 - 2x + 5x^2) / (1 - 3*x)^2.