LINKS
<a href="/index/Rec#order_05">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (1,98,-98,-1,1).
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Revision History for A245783
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
Numbers n such that the hexagonal number H(n) is equal to the sum of the pentagonal numbers P(m) and P(m+1) for some m.
(history; published version)
(history; published version)
STATUS
reviewed
approved
STATUS
proposed
reviewed
STATUS
editing
proposed
NAME
allocated Numbers n such that the hexagonal number H(n) is equal to the sum of the pentagonal numbers P(m) and P(m+1) for Colin Barkersome m.
DATA
1, 2, 57, 166, 5561, 16242, 544897, 1591526, 53394321, 155953282, 5232098537, 15281830086, 512692262281, 1497463395122, 50238609604977, 146736130891846, 4922871049025441, 14378643364005762, 482391124194888217, 1408960313541672806, 47269407300050019801
OFFSET
1,2
COMMENTS
Also nonnegative integers y in the solutions to 6*x^2-4*y^2+4*x+2*y+2 = 0, the corresponding values of x being A122513.
LINKS
Colin Barker, <a href="/A245783/b245783.txt">Table of n, a(n) for n = 1..1000</a>
<a href="/index/Rec#order_05">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,98,-98,-1,1).
FORMULA
a(n) = a(n-1)+98*a(n-2)-98*a(n-3)-a(n-4)+a(n-5).
G.f.: -x*(6*x^4+11*x^3-43*x^2+x+1) / ((x-1)*(x^2-10*x+1)*(x^2+10*x+1)).
EXAMPLE
57 is in the sequence because H(57) = 6441 = 3151+3290 = P(46)+P(47).
PROG
(PARI) Vec(-x*(6*x^4+11*x^3-43*x^2+x+1)/((x-1)*(x^2-10*x+1)*(x^2+10*x+1)) + O(x^100))
KEYWORD
allocated
nonn,easy
AUTHOR
Colin Barker, Dec 15 2014
STATUS
approved
editing
NAME
allocated for Colin Barker
KEYWORD
recycled
allocated
STATUS
editing
approved
NAME
Peaceable coexisting armies of queens: the maximum number m such that m white queens and m black queens can coexist on an n by n chessboard without attacking each other.
DATA
0, 0, 1, 2, 4, 5, 7, 9, 12, 14, 17, 21, 24
OFFSET
1,4
COMMENTS
28 <= a(14) <= 43, 32 <= a(15) <= 53, 37 <= a(16) <= 64, 42 <= a(17) <= 72, 47 <= a(18) <= 81, 52 <= a(19) <= 90, 58 <= a(20) <= 100. - Rob Pratt, Dec 01 2014
REFERENCES
Bosch, Robert A., "Peaceably coexisting armies of queens." Optima (Newsletter of the Mathematical Programming Society) 62.6-9 (1999): 271.
LINKS
Steven Prestwich and J. Christopher Beck, <a href="http://tidel.mie.utoronto.ca/pubs/pseudo.pdf">Exploiting Dominance in Three Symmetric Problems</a>, in: Fourth International Workshop on Symmetry and Constraint Satisfaction Problems (2004).
Barbara M. Smith, Karen E. Petrie, and Ian P. Gent, <a href="http://ipg.host.cs.st-andrews.ac.uk/papers/spgW9.pdf">Models and symmetry breaking for 'Peaceable armies of queens'</a>, Lecture Notes in Computer Science 3011 (2004), 271-286.
Barbara M. Smith, Karen E. Petrie, and Ian P. Gent, <a href="/A245783/a245783.pdf">Equal sized armies of queens on an 11x11 board</a> (Fig. 2 from the reference)
FORMULA
It is known that there is an asymptotic lower bound of (9/64)*n^2.
EXAMPLE
Unique solution (up to obvious symmetries) for n=3:
---
W..
...
.B.
---
A solution for n=4:
----
W..W
....
....
.BB.
----
One solution for n=5 puts one set of four queens in the corners and the other set in the squares a knight's move away:
-----
W...W
..B..
.B.B.
..B..
W...W
-----
There are two other solutions (up to symmetry) for n=5 (found by Rob Pratt, circa Sep 2014):
-----
..B.B
W....
..B.B
W....
.W.W.
-----
.W.W.
..W..
B...B
..W..
B...B
-----
A solution for n=6:
------
.WW...
..W..W
.....W
......
B...B.
B..BB.
------
A solution for n=12 (from Prestwich/Beck paper):
------------
...BBB.....B
...BBB....B.
...BBB...B.B
....B.....BB
.........BBB
.........BB.
..W...W.....
.WW.........
WWW.....W...
WW.....WW...
W.W...WWW...
.W....WWW...
------------
A solution for n=13 (from Prestwich/Beck paper):
-------------
B...B.B...B.B
..W.....W....
.W.W.W.W.W.W.
..W.....W....
B...B.B...B.B
..W.....W....
B...B.B...B.B
..W.....W....
.W.W.W.W.W.W.
..W.....W....
B...B.B...B.B
..W.....W....
B...B.B...B.B
-------------
KEYWORD
nonn,nice,more,changed
recycled
AUTHOR
Don Knuth, Aug 01 2014
EXTENSIONS
STATUS
approved
editing
STATUS
editing
approved
NAME
Peaceable coexisting armies of queens: the maximum number m such that m white queens and m black queens can coexist on an n X by n chessboard without attacking each other.
STATUS
proposed
editing