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.

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

Author

Jörg Zurkirchen, Apr 05 2014

Keywords

• nonn
• easy

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

Examples

			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\__/
.        \__/
.
		

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)
• Tim Crinion, Chamber Graphs of some geometries related to the Petersen Graph, 2013.
• 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
• Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Crossrefs

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

Programs

Magma

[n*(n-1)-Floor((n+1)/3): n in [1..60]]; // Vincenzo Librandi, Nov 12 2014

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 *)

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