A118201 Smallest difference such that both difference and number do not occur previously.

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 44, 68, 93, 66, 38, 67, 37, 5, 36, 69, 35, 70, 34, 71, 33, 72, 32, 73, 31, 74, 30, 75, 29, 76, 28, 77, 27, 78, 26, 79, 133, 188, 132, 189, 131, 190, 130, 191, 129, 192, 128, 193, 127, 60, 134

Offset: 0

Author

Franklin T. Adams-Watters, Apr 14 2006

Keywords

• nonn
• look
• nice

Comments

Similar to Recamán's sequence (A005132), but increases the difference to avoid duplicating values. Conjecture that every nonnegative integer occurs in this sequence. Evaluating through n=20000, the smallest number that has not occurred is 139.

The first 14 terms appear in the original OEIS logo. - Philippe Deléham, Mar 01 2013

This is very similar to A064389, and arguably just as nice. - Franklin T. Adams-Watters, Nov 11 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A118202 (inverse), A005132, A064389.

Programs

Maple

N:= 1000: # get all terms up to the first member > N
a[0]:= 0:
davail:= [$1..N]:
B:= Vector(2*N):
for n from 1 do
  found:= false;
  for i from 1 to nops(davail) do
    d:= davail[i];
    an:= a[n-1]-d;
    if an > 0 and B[an] = 0 then
      a[n]:= an; found:= true; break
    fi;
    ap:= a[n-1]+d;
    if B[ap] = 0 then
      a[n]:= ap; found:= true; break
    fi
  od:
  if (not found) or (a[n] > N) then break fi;
  davail:= subsop(i=NULL,davail);
  B[a[n]]:= 1;
od:
seq(a[i],i=0..n);  # Robert Israel, Nov 17 2015

Mathematica

M = 1000; (* get all terms up to the first member > M *)
a[0] = 0;
davail = Range[M];
B = Array[0&, 2M];
For[n = 1, True, n++,
found = False;
For[i = 1, i <= Length[davail], i++,
d = davail[[i]];
an = a[n-1] - d;
If[an > 0 && B[[an]] == 0,
a[n] = an; found = True; Break[]
];
ap = a[n-1] + d;
If[B[[ap]] == 0,
a[n] = ap; found = True; Break[]
]
];
If [Not @ found || (a[n] > M), Break[]];
davail = ReplacePart[davail, i -> Nothing];
B[[a[n]]] = 1;
];
Table[a[i], {i, 0, n}] (* Jean-François Alcover, Oct 24 2016, translated from Robert Israel's Maple code *)

Formula

a(n+1) = a(n)-d or a(n)+d, where a(n+1) must be positive and must not have occurred previously in the sequence; choose the smallest positive d such that this is possible where d is not |a(m+1)-a(m)| for any m < n; if both a(n)-d and a(n)+d have not occurred, choose a(n)-d.