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A008486
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Expansion of (1 + x + x^2)/(1 - x)^2.
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86
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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, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168, 171, 174, 177, 180, 183, 186
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OFFSET
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0,2
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COMMENTS
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Also the Engel expansion of exp^(1/3); cf. A006784 for the Engel expansion definition. - Benoit Cloitre, Mar 03 2002
Coordination sequence for planar net 6^3 (the graphite net, or the graphene crystal) - that is, the number of atoms at graph distance n from any fixed atom. Also for the hcb or honeycomb net. - N. J. A. Sloane, Jan 06 2013, Mar 31 2018
Coordination sequence for 2-dimensional cyclotomic lattice Z[zeta_3].
Conjecture: This is also the maximum number of edges possible in a planar simple graph with n+2 vertices. - Dmitry Kamenetsky, Jun 29 2008
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, Apr 23 2009
a(n) = sum of natural numbers m such that n - 1 <= m <= n + 1. Generalization: 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). - Jaroslav Krizek, Nov 18 2009
Integers n dividing a(n) = a(n-1) - a(n-2) with initial conditions a(0)=0, a(1)=1 (see A128834 with offset 0). - Thomas M. Bridge, Nov 03 2013
a(n) is conjectured to be the number of polygons added after n iterations of the polygon expansions (type A, B, C, D & E) shown in the Ngaokrajang link. The patterns are supposed to become the planar Archimedean net 3.3.3.3.3.3, 3.6.3.6, 3.12.12, 3.3.3.3.6 and 4.6.12 respectively when n - > infinity. - Kival Ngaokrajang, Dec 28 2014
Number of reduced words of length n in Coxeter group on 3 generators S_i with relations (S_i)^2 = (S_i S_j)^3 = I. - Ray Chandler, Nov 21 2016
Conjecture: let m = n + 2, p is the polyhedron formed by the convex hull of m points, q is the number of quadrilateral faces of p (see the Wikipedia link below), and f(m) = a(n) - q. Then f(m) would be the solution of the Thompson problem for all m in 3-space. - Sergey Pavlov, Feb 03 2017
Also, sequence defined by a(0)=1, a(1)=3, c(0)=2, c(1)=4; and thereafter a(n) = c(n-1) + c(n-2), and c consists of the numbers missing from a (see A001651). - Ivan Neretin, Mar 28 2017
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REFERENCES
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J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 158.
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LINKS
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Reticular Chemistry Structure Resource, hcb
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FORMULA
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a(0) = 1; a(n) = 3*n = A008585(n), n >= 1.
Euler transform of length 3 sequence [3, 0, -1]. - Michael Somos, Aug 04 2009
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EXAMPLE
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G.f. = 1 + 3*x + 6*x^2 + 9*x^3 + 12*x^4 + 15*x^5 + 18*x^6 + 21*x^7 + 24*x^8 + ...
Illustration of initial terms as triangles:
. o
. o o o
. o o o o o
. o o o o o o o
. o o o o o o o o o
. o o o o o o o o o o o o o o o o o o o o o
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. 1 3 6 9 12 15
(End)
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MATHEMATICA
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CoefficientList[Series[(1 + x + x^2) / (1 - x)^2, {x, 0, 80}], x] (* Vincenzo Librandi, Nov 23 2014 *)
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PROG
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(PARI) {a(n) = if( n==0, 1, 3 * n)}; /* Michael Somos, May 05 2015 */
(Haskell)
a008486 0 = 1; a008486 n = 3 * n
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CROSSREFS
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List of coordination sequences for Laves tilings (or duals of uniform planar nets): [3,3,3,3,3.3] = A008486; [3.3.3.3.6] = A298014, A298015, A298016; [3.3.3.4.4] = A298022, A298024; [3.3.4.3.4] = A008574, A296368; [3.6.3.6] = A298026, A298028; [3.4.6.4] = A298029, A298031, A298033; [3.12.12] = A019557, A298035; [4.4.4.4] = A008574; [4.6.12] = A298036, A298038, A298040; [4.8.8] = A022144, A234275; [6.6.6] = A008458.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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