A169677 The first of a pair of sequences A and B with property that all the differences |a_i - b_j| are distinct - for precise definition see Comments lines.

0, 1, 7, 18, 35, 59, 88, 125, 178, 233, 285, 344, 352, 442, 557, 675, 796, 797, 957, 1011, 1220, 1411, 1564, 1579, 1888, 2120, 2152, 2503, 2829, 2953, 3393, 3464, 3593, 3724, 4237, 4956, 5310, 5388, 5968, 6478, 6756, 7344, 7698, 8004, 8182

Offset: 1

Author

R. K. Guy and N. J. A. Sloane, Mar 27 2010

Keywords

• nonn
• nice

Comments

Consider pairs of sequences A = a_1 a_2 a_3 a_4 ... and B = b_1 b_2 b_3 ... such that

1: All the terms are nonnegative integers

2: The terms of A are strictly increasing

3: The terms of B are strictly increasing

4: All the numbers |a_i - b_j| are distinct

5: The terms are computed in the following order: a(1), b(1), a(2), b(2), ..., b(n-1), a(n), b(n), a(n+1), ... and always the smallest value is chosen that satisfies constraints 1-4.

Computed by Alois P. Heinz and Wouter Meeussen, Mar 27 2010

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A169678, A169679, A169680, A169690-A169693.

Programs

Maple

# Maple program from Alois P. Heinz:
ab:=proc() false end: ab(0):=true:
a:= proc(n) option remember;
local ok,i,k,s;
if n=1 then 0
else b(n-1);
for k from a(n-1)+1 do
ok:=true;
for i from 1 to n-1 do
if ab(abs(k-b(i))) then ok:= false; break fi
od;
if ok then s:={};
for i from 1 to n-1 do
s:= s union {abs(k-b(i))};
od
fi;
if ok and nops(s)=n-1 then break fi
od;
for i from 1 to n-1 do
ab(abs(k-b(i))):=true
od;
k
fi
end;
b:= proc(n) option remember;
local ok,i,k,s;
if n=1 then 0
else a(n);
for k from b(n-1)+1 do
ok:=true;
for i from 1 to n do
if ab(abs(k-a(i))) then ok:= false; break fi
od;
if ok then s:={};
for i from 1 to n do
s:= s union {abs(k-a(i))};
od
fi;
if ok and nops(s)=n then break fi
od;
for i from 1 to n do
ab(abs(k-a(i))):=true
od;
k
fi
end;
seq(a(n), n=1..80);
seq(b(n), n=1..80);

Mathematica

ClearAll[ab, a, b]; ab[] = False; ab[0] = True; a[n] := a[n] = Module[{ ok, i, k, s}, If[ n == 1 , 0, b[n-1]; For[ k = a[n-1] + 1 , True, k++, ok = True; For[ i = 1 , i <= n-1, i++, If[ ab[Abs[k - b[i]]] , ok = False; Break[] ]]; If[ ok , s = {}; For[ i=1 , i <= n-1 , i++, s = s ~Union~ {Abs[k - b[i]]};]]; If[ ok && (Length[s] == n-1) , Break[] ]]; For[ i=1 , i <= n-1 , i++, ab[Abs[k - b[i]]] = True]; k]]; b[n_] := b[n] = Module[{ ok, i, k, s}, If[ n == 1 , 0, a[n]; For[ k = b[n-1] + 1 , True, k++, ok = True; For[ i=1 , i <= n, i++, If[ ab[Abs[k - a[i]]] , ok = False; Break[] ]]; If[ ok , s = {}; For[ i=1 , i <= n , i++, s = s ~Union~ {Abs[k - a[i]]};]]; If[ ok && Length[s] == n , Break[] ]]; For[ i=1 , i <= n, i++, ab[Abs[k - a[i]]] := True]; k]]; Table[a[n], {n, 1, 45}] (* Jean-François Alcover, Aug 13 2012, translated from Alois P. Heinz's Maple program *)

Extensions

Comments clarified by Zak Seidov and Alois P. Heinz, Apr 13 2010.