|
| |
|
|
A334559
|
|
a(2n+1) = (2n)^2 - 1, a(4n+2) = (4n+2)^2 - 3, a(4n) = (4n)^2 - 5, except for a(1) = 1, a(4) = 2. (Conjectured outcome of an elimination game, see comments.)
|
|
1
|
|
|
|
1, 1, 3, 2, 15, 33, 35, 59, 63, 97, 99, 139, 143, 193, 195, 251, 255, 321, 323, 395, 399, 481, 483, 571, 575, 673, 675, 779, 783, 897, 899, 1019, 1023, 1153, 1155, 1291, 1295, 1441, 1443, 1595, 1599, 1761, 1763, 1931, 1935, 2113, 2115, 2299, 2303, 2497, 2499
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
Also (conjectured) the "winner" or largest remaining number in the following game:
An n X n board is initialized with n^2 chess kings, numbered in raster order. The kings retain their labels after they move. They take turns moving; after king "n^2" moves, king 1 gets to move again.
If a king has already been captured, or has no captures available to it, it passes its turn. On each turn, a king captures the highest-labeled king available. Play continues until no remaining king can capture. The winner is the highest-numbered king remaining. - Allan C. Wechsler, May 01 2020
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: x*(1 + x^2 - x^3 + 10*x^4 + 19*x^5 - 12*x^6 + 7*x^7 - 9*x^8 - 9*x^9 + 9*x^10) / ((1 - x)^3*(1 + x)^2*(1 + x^2)).
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7) for n>11.
(End)
|
|
|
EXAMPLE
|
For n=3 we get the following steps:
1-2-3 0-2-3 0-0-3 0-0-0 0-0-0 0-0-0 0-0-0 0-0-0 0-0-0
4-5-6 -> 4-1-6 -> 4-1-2 -> 4-1-3 -> 0-1-3 -> 0-1-3 -> 0-1-3 -> 0-0-3 -> 0-0-0
7-8-9 7-8-9 7-8-9 7-8-9 7-4-9 0-7-9 0-9-0 0-1-0 0-3-0
Therefore a(3)=3, the highest remaining number.
|
|
|
PROG
|
(PARI) kings_battle(n)={my( N=n^2, K=[1..N] /*position of king i*/, board=K /* king on field i */, i=N, coord(F)=divrem(F-1, n)+[1, 1]~, done); until( done, my( last=i ); until( i == last && done = 1 /*tried all kings*/, K[ i = i%N+1 ] || next /*already taken*/; my( xy=coord(K[i]), m=0); /* see whether this king can take any other one */ forvec( d=vectorv(2, i, [-(xy[i]>1), xy[i]<n]), d && board[ K[i] + [n, 1]*d ]>m && m = board[ K[i] + [n, 1]*d ] ); m || next; /* we can take king m */; board[ K[i] ] = 0/*empty*/; K[i] = K[m]; K[m]=0/*dead*/; board[ K[i]] = i; next(2))); i=vecmax(board); [i, coord(K[i]), board] } \\ for illustration
(PARI) apply( A334559(n)={if(bittest(n, 0), if(n>1, (n-1)^2-1, 1), n==4, 2, n^2-5+bitand(n, 2))}, [1..66]) \\ M. F. Hasler, May 22 2020
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
