A307797 Lexicographically earliest version of a self referencing "Kimberling shuffle" expulsion array sequence.

1, 2, 2, 3, 2, 4, 4, 5, 2, 2, 6, 2, 7, 8, 4, 4, 2, 6, 9, 2, 10, 4, 2, 2, 2, 11, 12, 4, 6, 11, 7, 6, 9, 9, 6, 2, 11, 11, 7, 13, 11, 2, 2, 7, 14, 2, 11, 9, 15, 16, 17, 17, 10, 7, 11, 2, 18, 11, 4, 7, 2, 12, 19, 18, 11, 4, 10, 11, 10, 6, 11, 13, 20, 21, 14, 6, 22, 23, 2, 24, 4, 15, 2, 2, 25, 20, 26, 12, 27, 7, 16, 28, 29, 30, 11, 31, 13, 29, 11

Offset

1,2

Comments

Start with this sequence, "shuffle" as in A007063 and the sequence reappears in the diagonal of the array. Terms are transformed from A307536 to this lexicofirst version by replacing the first and all subsequent occurrences of any term > all preceding terms by k+1, where k is the greatest (transformed) term seen so far. The records of this sequence is the natural numbers, A000027, starting point of the original Kimberling exclusion array.

Links

Table of n, a(n) for n=1..99.

D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998.

C. Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, pp. 82-83.

Example

A307536(4)=4 > all preceding terms, the greatest of which is 2, so a(4)=3. Since 4 appears only once in A307536, 3 appears only once in this sequence.
A307536(21)=21 > all preceding terms, the greatest of which (in this sequence) is 9, so a(21)=10. Subsequent terms with the same value are a(53), a(67), a(69), ... because the corresponding terms (same indices) in A307536 all have value 21.

Prog

(PARI)
A(z) = {x=z; y=z; xx=2*x-4; while (y<=xx, x--; xx-=2; if (bittest(y, 0)==1, y=x+((y+1)>>1), y=x-(y>>1))); return(x+y-1); } \\ A007063
B(z) = {a=z; n=1; while (a!=n, if (a<n, a=2*(n-a), a>2*n, a--, a=2*(a-n)-1); n++); return(a); } \\ A006852
addgroup(group, n, fixed, v) = {my(ok = 1, m=v[n]); while(ok, listput(group, m); if (m==n, ok=0; break); if (m > #v, ok=0; break); n = m; m = v[n]; ); group; }
makegroup(n, fixed, va, vb) = {my(group = List()); listput(group, n); group = addgroup(group, n, fixed, va); group = addgroup(group, n, fixed, vb); listsort(group, 1); Vec(group); }
setgroup(v, n, group) = {my(gmin = vecmin(group)); for (i=1, #group, if ((group[i] <= #v) && !v[n], v[n] = gmin); ); v; }
lista() = {nn = 200; nout = 90; va = vector(nn, k, A(k)); vb = vector(nn, k, B(k)); vc = vector(nn); fixed = List(); for (n = 1, nn, if (va[n] == n, listput(fixed, n)); ); fixed = Vec(fixed); for (n=1, nn, group = makegroup(n, fixed, va, vb); vc = setgroup(vc, n, group); ); vector(nout, k, vc[k]); } \\ A307536
earliest(v) = {my(m = Map(), val=1); for (i=1, #v, if (!mapisdefined(m, v[i]), mapput(m, v[i], val); val++); ); apply(x->mapget(m, x), v); }
earliest(lista()) \\ Michel Marcus, Jun 14 2019

Crossrefs

Cf. A007063, A307536, A006852, A000027.

Sequence in context: A106289 A332436 A165418 * A233777 A048620 A291271

Adjacent sequences: A307794 A307795 A307796 * A307798 A307799 A307800

Keyword

nonn

Author

David James Sycamore and Lars Blomberg, Apr 29 2019

Status

approved