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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.

%I A240438 #67 Sep 08 2022 08:46:07
%S A240438 0,1,5,11,18,28,40,53,69,87,106,128,152,177,205,235,266,300,336,373,
%T A240438 413,455,498,544,592,641,693,747,802,860,920,981,1045,1111,1178,1248,
%U A240438 1320,1393,1469,1547,1626,1708,1792,1877,1965,2055,2146,2240,2336,2433,2533,2635
%N 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.
%C A240438 Difference table of a(n), with a(0)=0 and offset=0:
%C A240438 0,   0,  1,  5, 11, 18, 28, 40, 53, 69, ...
%C A240438 0,   1,  4,  6,  7, 10, 12, 13, 16, 18, ...   =  A047234(n+1)
%C A240438 1,   3,  2,  1,  3,  2,  1,  3,  2,  1, ...   =  A130784
%C A240438 2,  -1, -1,  2, -1, -1,  2, -1, -1,  2, ...   = -A131713(n+1)
%C A240438 -3,  0,  3, -3,  0,  3, -3,  0,  3, -3; ...   =  A099838(n+4)
%C A240438 3,   3, -6,  3,  3, -6,  3,  3, -6,  3, ...
%C A240438 0,  -9,  9,  0, -9,  9,  0, -9,  9,  0, ...
%C A240438 -9, 18, -9, -9, 18, -9, -9, 18, -9, -9, ...
%C A240438 First column: see A057682. - _Paul Curtz_, Nov 11 2014
%C A240438 Diameter of the chamber graph Γ(Alt(2n+1)). Definition of this graph:
%C A240438 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 A240438 Vertices u and v are connected iff either:
%C A240438 u and v are identical except for their first elements u[1] and v[1], or
%C A240438 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 A240438 22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»
%H A240438 Vincenzo Librandi, <a href="/A240438/b240438.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Jörg Zurkirchen)
%H A240438 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 A240438 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 A240438 Jörg Zurkirchen, <a href="/A240438/a240438_1.pdf">Honeycomb.pdf</a>
%H A240438 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1, 1, -2, 1).
%F A240438 a(n) = n*(n-1)-floor((n+1)/3).
%F A240438 G.f.: -x^2*(x+1)*(2*x+1) / ((x-1)^3*(x^2+x+1)). - _Colin Barker_, Apr 08 2014
%F A240438 a(n+3) = a(n) + 6*n+5. - _Paul Curtz_, Nov 11 2014
%F A240438 a(n) = n^2 - (A042965(n+1)=0, 1, 3, 4, ...). - _Paul Curtz_, Nov 11 2014
%F A240438 a(n+1) = a(n) + A047234(n+1). - _Paul Curtz_, Nov 11 2014
%e A240438 For n = 3 an example of a honeycomb with the greatest minimal difference of a(3) = 5 is:
%e A240438 .         __
%e A240438 .      __/ 7\__
%e A240438 .   __/15\__/13\__
%e A240438 .  / 4\__/ 2\__/ 1\
%e A240438 .  \__/10\__/ 8\__/
%e A240438 .  /18\__/16\__/14\
%e A240438 .  \__/ 5\__/ 3\__/
%e A240438 .  /12\__/11\__/ 9\
%e A240438 .  \__/19\__/17\__/
%e A240438 .     \__/ 6\__/
%e A240438 .        \__/
%e A240438 .
%p A240438 A240438:=n->n*(n-1)-floor((n+1)/3); seq(A240438(n), n=1..50); # _Wesley Ivan Hurt_, Apr 08 2014
%t A240438 Table[n (n - 1) - Floor[(n + 1)/3], {n, 50}] (* _Wesley Ivan Hurt_, Apr 08 2014 *)
%t A240438 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 A240438 LinearRecurrence[{2, -1, 1, -2, 1},{0, 1, 5, 11, 18},52] (* _Ray Chandler_, Sep 24 2015 *)
%o A240438 (Magma) [n*(n-1)-Floor((n+1)/3): n in [1..60]]; // _Vincenzo Librandi_, Nov 12 2014
%Y A240438 Cf. A042965, A047234, A057682, A099838, A130784, A131713.
%K A240438 nonn,easy
%O A240438 1,3
%A A240438 _Jörg Zurkirchen_, Apr 05 2014