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, 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

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

Table of n, a(n) for n=1..51.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,-1,1).

Formula

From Colin Barker, May 06 2020: (Start)

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

Sequence in context: A133932 A111999 A286947 * A190961 A346059 A126323

Adjacent sequences: A334556 A334557 A334558 * A334560 A334561 A334562

Keyword

nonn

Author

Ali Sada and M. F. Hasler, May 06 2020

Status

approved