0,2

Also the Engel expansion of exp^(1/3); cf. A006784 for the Engel expansion definition - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 03 2002

Coordination sequence for graphite net.

Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_3].

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.

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

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]

J.-G. Eon, Algebraic determination of generating functions for coordination sequences in crystal structures, Acta Cryst. A58 (2002), 47-53.

M. Beck and S. Hosten, Cyclotomic polytopes and growth series of cyclotomic lattices, arXiv math.CO/0508136.

A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004. (See Table 5.)

a(0) = 1; a(n) = 3n, n >= 1.

Euler transform of length 3 sequence [ 3, 0, -1]. - Michael Somos Aug 04 2009

1 + 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + 24*x^8 + ...

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]

(PARI) {a(n) = if( n<1, n==0, 3 * n)} /* Michael Somos Aug 04 2009 */

Sequence in context: A161351 A008585 A031193 this_sequence A135943 A036686 A059563

Adjacent sequences: A008483 A008484 A008485 this_sequence A008487 A008488 A008489

nonn,new

N. J. A. Sloane (njas(AT)research.att.com).

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