Search: id:A008486
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%I A008486
%S A008486 1,3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,
%T A008486 75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,129,132,
%U A008486 135,138,141,144,147,150,153,156,159,162,165,168,171,174,177,180,183,186
%N A008486 Expansion of (1+x+x^2)/(1-x)^2.
%C A008486 Also the Engel expansion of exp^(1/3); cf. A006784 for the Engel expansion 
               definition - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 03 2002
%C A008486 Coordination sequence for graphite net.
%C A008486 Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_3].
%C A008486 Conjecture from Dmitry Kamenetsky (dmitry.kamenetsky(AT)rsise.anu.edu.au), 
               Jun 29 2008: This is also the maximum number of edges possible in 
               a planar simple graph with n+2 vertices.
%C A008486 The conjecture is correct. Proof: For n=0 the theorem holds, the maximum 
               planar graph has n+2=2 vertices and 1 edge. Now suppose that we have 
               a connected planar graph with at least 3 vertices. If it contains 
               a face that is not a triangle, we can add an edge that divides this 
               face into two without breaking its planarity. Hence all maximum planar 
               graphs are triangulations. Euler's formula for planar graphs states 
               that in any planar simple graph with V vertices, E edges and F faces 
               we have V+F-E=2. If all faces are triangles, then F=2E/3, which gives 
               us E=3V-6. Hence for n>0 each maximum planar simple graph with n+2 
               vertices has 3n edges. - Michal Forisek (misof(AT)oeis.ksp.sk), Apr 
               23 2009
%C A008486 a(n) = sum of natural numbers m such that n - 1 <= m <= n + 1. Generalisation: 
               If a(n,k) = sum of natural numbers m such that n - k <= m <= n + 
               k (k >= 1) then a(n,k) = (k + n)*(k + n + 1)/2 = A000217(k+n) for 
               0 <= n <= k, a(n,k) = a(n-1,k) +2k + 1 = ((k + n - 1)*(k + n)/2) 
               + 2k + 1 = A000217(k+n-1) +2k +1 for n >= k + 1 (see e.g. A008486). 
               [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Nov 18 2009]
%D A008486 J.-G. Eon, Algebraic determination of generating functions for coordination 
               sequences in crystal structures, Acta Cryst. A58 (2002), 47-53.
%D A008486 M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic 
               lattices, arXiv math.CO/0508136.
%H A008486 A. S. Fraenkel, <a href="http://www.integers-ejcnt.org/">New games related 
               to old and new sequences</a>, INTEGERS, Electronic J. of Combinatorial 
               Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)
%F A008486 a(0) = 1; a(n) = 3n, n >= 1.
%F A008486 Euler transform of length 3 sequence [ 3, 0, -1]. - Michael Somos Aug 
               04 2009
%e A008486 1 + 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + 24*x^8 
               + ...
%e A008486 a(n) = (1 + n)*(2 + n)/2 = A000217(1+n) for 0 <= n <= 1, a(n) = a(n-1) 
               + 3 for n >= 2. [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), 
               Nov 18 2009]
%o A008486 (PARI) {a(n) = if( n<1, n==0, 3 * n)} /* Michael Somos Aug 04 2009 */
%Y A008486 Sequence in context: A161351 A008585 A031193 this_sequence A135943 A036686 
               A059563
%Y A008486 Adjacent sequences: A008483 A008484 A008485 this_sequence A008487 A008488 
               A008489
%K A008486 nonn,new
%O A008486 0,2
%A A008486 N. J. A. Sloane (njas(AT)research.att.com).
%E A008486 The conjecture was true, I provided a proof and fixed an off-by-two error 
               Michal Forisek (misof(AT)oeis.ksp.sk), Apr 22 2009

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