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A240438
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Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.
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6
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0, 1, 5, 11, 18, 28, 40, 53, 69, 87, 106, 128, 152, 177, 205, 235, 266, 300, 336, 373, 413, 455, 498, 544, 592, 641, 693, 747, 802, 860, 920, 981, 1045, 1111, 1178, 1248, 1320, 1393, 1469, 1547, 1626, 1708, 1792, 1877, 1965, 2055, 2146, 2240, 2336, 2433, 2533, 2635
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OFFSET
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1,3
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COMMENTS
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Difference table of a(n), with a(0)=0 and offset=0:
0, 0, 1, 5, 11, 18, 28, 40, 53, 69, ...
0, 1, 4, 6, 7, 10, 12, 13, 16, 18, ... = A047234(n+1)
1, 3, 2, 1, 3, 2, 1, 3, 2, 1, ... = A130784
2, -1, -1, 2, -1, -1, 2, -1, -1, 2, ... = -A131713(n+1)
-3, 0, 3, -3, 0, 3, -3, 0, 3, -3; ... = A099838(n+4)
3, 3, -6, 3, 3, -6, 3, 3, -6, 3, ...
0, -9, 9, 0, -9, 9, 0, -9, 9, 0, ...
-9, 18, -9, -9, 18, -9, -9, 18, -9, -9, ...
Diameter of the chamber graph Γ(Alt(2n+1)). Definition of this graph:
Each vertex v is a sequence (v[1],v[2],...,v[n]) of length n, where each v[i] is a 2-subset of {1,2,...,2n+1} and v[i] and v[j] are disjoint unless i=j.
Vertices u and v are connected iff either:
u and v are identical except for their first elements u[1] and v[1], or
u and v are identical except for some i for which u[i]=v[i+1] and v[i]=u[i+1] - Tim Crinion, 17 Feb 2019
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REFERENCES
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22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»
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LINKS
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FORMULA
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a(n) = n*(n-1)-floor((n+1)/3).
G.f.: -x^2*(x+1)*(2*x+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Apr 08 2014
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EXAMPLE
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For n = 3 an example of a honeycomb with the greatest minimal difference of a(3) = 5 is:
. __
. __/ 7\__
. __/15\__/13\__
. / 4\__/ 2\__/ 1\
. \__/10\__/ 8\__/
. /18\__/16\__/14\
. \__/ 5\__/ 3\__/
. /12\__/11\__/ 9\
. \__/19\__/17\__/
. \__/ 6\__/
. \__/
.
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MAPLE
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MATHEMATICA
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Table[n (n - 1) - Floor[(n + 1)/3], {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[x (x + 1) (2 x + 1) / ((1 - x)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2014 *)
LinearRecurrence[{2, -1, 1, -2, 1}, {0, 1, 5, 11, 18}, 52] (* Ray Chandler, Sep 24 2015 *)
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PROG
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CROSSREFS
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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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