A167667 - OEIS

%I #35 Feb 16 2025 08:33:11

%S 1,3,12,36,96,240,576,1344,3072,6912,15360,33792,73728,159744,344064,

%T 737280,1572864,3342336,7077888,14942208,31457280,66060288,138412032,

%U 289406976,603979776,1258291200,2617245696,5435817984,11274289152,23353884672,48318382080

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

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

%H Vincenzo Librandi, <a href="/A167667/b167667.txt">Table of n, a(n) for n = 0..3000</a>

%H Milan Janjić, <a href="https://arxiv.org/abs/1905.04465">On Restricted Ternary Words and Insets</a>, arXiv:1905.04465 [math.CO], 2019.

%H Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Cube-ConnectedCycleGraph.html">Cube-Connected Cycle Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximalClique.html">Maximal Clique</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MaximumClique.html">Maximum Clique</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4).

%F a(0)=1, a(n) = 3*n*2^(n-1) for n>0.

%F a(0)=1, a(1)=3, a(2)=12, a(n) = 4*a(n-1)-4*a(n-2) for n>2.

%F a(n) = Sum_{k=0..n} A167666(n,k) * 2^k.

%F 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

%F a(0)=1, a(n) = Sum_{i=0..n} binomial(n,i) * (2n-i). - _Wesley Ivan Hurt_, Mar 20 2015

%p A167667:=n->3*n*2^(n-1): (1,seq(A167667(n), n=1..30)); # _Wesley Ivan Hurt_, Mar 20 2015

%t CoefficientList[Series[(1 - x + 4*x^2)/(1 - 2*x)^2, {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Mar 20 2015 *)

%t Join[{1}, LinearRecurrence[{4, -4}, {3, 12}, 20]] (* _Eric W. Weisstein_, Dec 01 2017 *)

%t Join[{1}, Table[3 2^(n - 1) n, {n, 20}]] (* _Eric W. Weisstein_, Dec 01 2017 *)

%t CoefficientList[Series[(1 - x + 4 x^2)/(-1 + 2 x)^2, {x, 0, 20}], x] (* _Eric W. Weisstein_, Dec 01 2017 *)

%o (PARI) Vec((1-x+4*x^2)/(1-2*x)^2 + O(x^50)) \\ _Michel Marcus_, Mar 21 2015

%o (PARI) a(n) = if(n==0, 1, 3*n*2^(n-1)); \\ _Altug Alkan_, May 16 2018

%o (Magma) [1] cat [3*n*2^(n-1): n in [1..30]]; // _Vincenzo Librandi_, Mar 21 2015

%Y Cf. A167666.

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Nov 08 2009