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A240438
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.
6
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
OFFSET
1,3
COMMENTS
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, ...
First column: see A057682. - Paul Curtz, Nov 11 2014
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
REFERENCES
22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 100 terms from Jörg Zurkirchen)
Fédération Suisse des Jeux Mathématiques, 22nd Championship of Mathematical and Logical Games - Quarter Final 2008, 18 problems in French; problem number 18 was decisive to creating this sequence. See following pdf for an English version of problem 18.
Jörg Zurkirchen, Honeycomb.pdf
FORMULA
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
a(n+3) = a(n) + 6*n+5. - Paul Curtz, Nov 11 2014
a(n) = n^2 - (A042965(n+1)=0, 1, 3, 4, ...). - Paul Curtz, Nov 11 2014
a(n+1) = a(n) + A047234(n+1). - Paul Curtz, Nov 11 2014
EXAMPLE
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\__/
. \__/
.
MAPLE
A240438:=n->n*(n-1)-floor((n+1)/3); seq(A240438(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014
MATHEMATICA
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 *)
PROG
(Magma) [n*(n-1)-Floor((n+1)/3): n in [1..60]]; // Vincenzo Librandi, Nov 12 2014
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jörg Zurkirchen, Apr 05 2014
STATUS
approved