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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.
6

%I #67 Sep 08 2022 08:46:07

%S 0,1,5,11,18,28,40,53,69,87,106,128,152,177,205,235,266,300,336,373,

%T 413,455,498,544,592,641,693,747,802,860,920,981,1045,1111,1178,1248,

%U 1320,1393,1469,1547,1626,1708,1792,1877,1965,2055,2146,2240,2336,2433,2533,2635

%N 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.

%C Difference table of a(n), with a(0)=0 and offset=0:

%C 0, 0, 1, 5, 11, 18, 28, 40, 53, 69, ...

%C 0, 1, 4, 6, 7, 10, 12, 13, 16, 18, ... = A047234(n+1)

%C 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, ... = A130784

%C 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, ... = -A131713(n+1)

%C -3, 0, 3, -3, 0, 3, -3, 0, 3, -3; ... = A099838(n+4)

%C 3, 3, -6, 3, 3, -6, 3, 3, -6, 3, ...

%C 0, -9, 9, 0, -9, 9, 0, -9, 9, 0, ...

%C -9, 18, -9, -9, 18, -9, -9, 18, -9, -9, ...

%C First column: see A057682. - _Paul Curtz_, Nov 11 2014

%C Diameter of the chamber graph Γ(Alt(2n+1)). Definition of this graph:

%C 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.

%C Vertices u and v are connected iff either:

%C u and v are identical except for their first elements u[1] and v[1], or

%C 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

%D 22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»

%H Vincenzo Librandi, <a href="/A240438/b240438.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Jörg Zurkirchen)

%H Tim Crinion, <a href="http://eprints.maths.manchester.ac.uk/2085/1/Chamber_graphs_of_some_geometries_related_to_the_Petersen_graph.PDF">Chamber Graphs of some geometries related to the Petersen Graph</a>, 2013.

%H Fédération Suisse des Jeux Mathématiques, <a href="http://homepage.hispeed.ch/FSJM/documents/22_Quarts_ind.pdf">22nd Championship of Mathematical and Logical Games - Quarter Final 2008</a>, 18 problems in French; problem number 18 was decisive to creating this sequence. See following pdf for an English version of problem 18.

%H Jörg Zurkirchen, <a href="/A240438/a240438_1.pdf">Honeycomb.pdf</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 1, -2, 1).

%F a(n) = n*(n-1)-floor((n+1)/3).

%F G.f.: -x^2*(x+1)*(2*x+1) / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Apr 08 2014

%F a(n+3) = a(n) + 6*n+5. - _Paul Curtz_, Nov 11 2014

%F a(n) = n^2 - (A042965(n+1)=0, 1, 3, 4, ...). - _Paul Curtz_, Nov 11 2014

%F a(n+1) = a(n) + A047234(n+1). - _Paul Curtz_, Nov 11 2014

%e For n = 3 an example of a honeycomb with the greatest minimal difference of a(3) = 5 is:

%e . __

%e . __/ 7\__

%e . __/15\__/13\__

%e . / 4\__/ 2\__/ 1\

%e . \__/10\__/ 8\__/

%e . /18\__/16\__/14\

%e . \__/ 5\__/ 3\__/

%e . /12\__/11\__/ 9\

%e . \__/19\__/17\__/

%e . \__/ 6\__/

%e . \__/

%e .

%p A240438:=n->n*(n-1)-floor((n+1)/3); seq(A240438(n), n=1..50); # _Wesley Ivan Hurt_, Apr 08 2014

%t Table[n (n - 1) - Floor[(n + 1)/3], {n, 50}] (* _Wesley Ivan Hurt_, Apr 08 2014 *)

%t CoefficientList[Series[x (x + 1) (2 x + 1) / ((1 - x)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* _Vincenzo Librandi_, Nov 12 2014 *)

%t LinearRecurrence[{2, -1, 1, -2, 1},{0, 1, 5, 11, 18},52] (* _Ray Chandler_, Sep 24 2015 *)

%o (Magma) [n*(n-1)-Floor((n+1)/3): n in [1..60]]; // _Vincenzo Librandi_, Nov 12 2014

%Y Cf. A042965, A047234, A057682, A099838, A130784, A131713.

%K nonn,easy

%O 1,3

%A _Jörg Zurkirchen_, Apr 05 2014