

A065188


"Greedy Queens" permutation of the natural numbers.


29



1, 3, 5, 2, 4, 9, 11, 13, 15, 6, 8, 19, 7, 22, 10, 25, 27, 29, 31, 12, 14, 35, 37, 39, 41, 16, 18, 45, 17, 48, 20, 51, 53, 21, 56, 58, 60, 23, 63, 24, 66, 28, 26, 70, 72, 74, 76, 78, 30, 32, 82, 84, 86, 33, 89, 34, 92, 38, 36, 96, 98, 100, 102, 40, 105, 107, 42, 110, 43, 113
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OFFSET

1,2


COMMENTS

This permutation is produced by a simple greedy algorithm: starting from the top left corner, walk along each successive antidiagonal of an infinite chessboard and place a queen in the first available position where it is not threatened by any of the existing queens. In other words, this permutation satisfies the condition that p(i+d) <> p(i)+d for all i and d >= 1.
p(n) = k means that a queen appears in column n in row k.  N. J. A. Sloane, Aug 18 2016
That this is a permutation follows from the proof in A269526 that every row and every column in that array is a permutation of the positive integers. In particular, every row and every column contains a 1 (which translates to a queen in the present sequence).  N. J. A. Sloane, Dec 10 2017
The graph of this sequence shows two straight lines of respective slope equal to the Golden Ratio A001622, Phi = 1+phi = (sqrt(5)+1)/2 and phi = 1/Phi = (sqrt(5)1)/2.  M. F. Hasler, Jan 13 2018
One has a(42) = 28 and a(43) = 26. Such irregularities make it difficult to get an explicit formula. They would not occur if the squares on the antidiagonals had been checked for possible positions starting from the opposite end, so as to ensure that the subsequences corresponding to the points on either line would both be increasing. Then one would have that a(n1) is either round(n*phi)+1 or round(n/phi)+1. (The +1's could all be avoided if the origin were taken as a(0) = 0 instead of a(1) = 1.) Presently most values are such that either round(n*phi) or round(n/phi) does not differ by more than 1 from a(n1)1, except for very few exceptions of the above form (a(42) being the first of these).  M. F. Hasler, Jan 15 2018


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: nonattacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Matteo Fischetti, Domenico Salvagnin, Chasing First Queens by Integer Programming, 2018.
Matteo Fischetti, Domenico Salvagnin, Finding First and MostBeautiful Queens by Integer Programming, arXiv:1907.08246 [cs.DS], 2019.
N. J. A. Sloane, Scatterplot of first 100 terms
N. J. A. Sloane, Table of n, a(n) for n = 1..50000 [Obtained using the Maple program of Alois P. Heinz]
Index entries for sequences that are permutations of the natural numbers


FORMULA

It would be nice to have a formula!  N. J. A. Sloane, Jun 30 2016


EXAMPLE

The top left corner of the board is:
+
 Q x x x x x x x x x ...
 x x x Q x x x x x x ...
 x Q x x x x x x x x ...
 x x x x Q x x x x x ...
 x x Q x x x x x x x ...
 x x x x x x x x x Q ...
 x x x x x x x x x x ...
 x x x x x x x x x x ...
 x x x x x Q x x x x ...
 ...
which illustrates p(1)=1, p(2)=3, p(3)=5, p(4)=2, etc.  N. J. A. Sloane, Aug 18 2016, corrected Aug 21 2016


MAPLE

# Maple program from Alois P. Heinz, Aug 19 2016: (Start)
# For more terms change the constant 100.
max_diagonal:= 3* 100: # make this large enough 
# about 3*max number of terms
h:= proc() true end: # horizontal line free?
v:= proc() true end: # vertical line free?
u:= proc() true end: # up diagonal free?
d:= proc() true end: # down diagonal free?
a:= proc() 0 end: # for A065188
b:= proc() 0 end: # for A065189
for t from 2 to max_diagonal do
if u(t) then
for j to t1 do
i:= tj;
if v(j) and h(i) and d(ij) then
v(j), h(i), d(ij), u(i+j):= false$4;
a(j):= i;
b(i):= j;
break
fi
od
fi
od:
seq(a(n), n=1..100); # this is A065188
seq(b(n), n=1..100); # this is A065189 # (End)
GreedyQueens(256); GreedyQueens := upto_n > PM2PL(GreedyNonThreateningPermutation(upto_n, 1, 1), upto_n);
SquareThreatened := proc(a, i, j, upto_n, senw, nesw) local k; for k from 1 to i do if(a[k, j] > 0) then RETURN(1); fi; od; for k from 1 to j do if(a[i, k] > 0) then RETURN(1); fi; od; if((1 = i) and (1 = j)) then RETURN(0); fi; for k from 1 to `if`((1 = senw), min(i, j)1, senw) do if(a[ik, jk] > 0) then RETURN(1); fi; od; for k from 1 to `if`((1 = nesw), i1, nesw) do if(a[ik, j+k] > 0) then RETURN(1); fi; od; for k from 1 to `if`((1 = nesw), j1, nesw) do if(a[i+k, jk] > 0) then RETURN(1); fi; od; RETURN(0); end;
GreedyNonThreateningPermutation := proc(upto_n, senw, nesw) local a, i, j; a := array(1..upto_n, 1..upto_n); for i from 1 to upto_n do for j from 1 to upto_n do a[i, j] := 0; od; od; for j from 1 to upto_n do for i from 1 to j do if(0 = SquareThreatened(a, i, (ji+1), upto_n, senw, nesw)) then a[i, ji+1] := 1; fi; od; od; RETURN(eval(a)); end;
PM2PL := proc(a, upto_n) local b, i, j; b := []; for i from 1 to upto_n do for j from 1 to upto_n do if(a[i, j] > 0) then break; fi; od; b := [op(b), `if`((j > upto_n), 0, j)]; od; RETURN(b); end;


MATHEMATICA

Fold[Function[{a, n}, Append[a, 2 + LengthWhile[Differences@ Union@ Apply[Join, MapIndexed[Select[#2 + #1 {1, 0, 1}, # > 0 &] & @@ {n  First@ #2, #1} &, a]], # == 1 &]]], {1}, Range[2, 70]] (* Michael De Vlieger, Jan 14 2018 *)


CROSSREFS

A065185 gives the associated p(i)i delta sequence. A065186 gives the corresponding permutation for "promoted rooks" used in Shogi, A065257 gives "Quintal Queens" permutation.
A065189 gives inverse permutation.
See A199134, A275884, A275890, A275891, A275892 for information about the split of points below and above the diagonal.
Cf. A269526.
If we subtract 1 and change the offset to 0 we get A275895, A275896, A275893, A275894.
Tracking at which squares along the successive antidiagonals the queens appear gives A275897 and A275898.
Antidiagonal and diagonal indices give A276324 and A276325.
Sequence in context: A109313 A331526 A176881 * A065257 A258428 A090396
Adjacent sequences: A065185 A065186 A065187 * A065189 A065190 A065191


KEYWORD

nonn


AUTHOR

Antti Karttunen, Oct 19 2001


STATUS

approved



