A167667 Expansion of (1-x+4*x^2)/(1-2*x)^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
Links
- 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).
Crossrefs
Cf. A167666.
Programs
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Magma
[1] cat [3*n*2^(n-1): n in [1..30]]; // Vincenzo Librandi, Mar 21 2015
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Maple
A167667:=n->3*n*2^(n-1): (1,seq(A167667(n), n=1..30)); # Wesley Ivan Hurt, Mar 20 2015
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Mathematica
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 *) -
PARI
Vec((1-x+4*x^2)/(1-2*x)^2 + O(x^50)) \\ Michel Marcus, Mar 21 2015
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PARI
a(n) = if(n==0, 1, 3*n*2^(n-1)); \\ Altug Alkan, May 16 2018
Formula
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
Comments