A284917 Lexicographically earliest infinite sequence of positive integers such that the Taxicab distance is unique to each pair of distinct points ((n, a(n)), (k, a(k)).

1, 1, 2, 5, 9, 16, 25, 38, 58, 72, 87, 112, 136, 169, 190, 237, 274, 344, 383, 456, 545, 572, 640, 752, 798, 891, 944, 989, 1131, 1283, 1365, 1492, 1540, 1788, 1862, 1994, 2218, 2342, 2472, 2741, 2885, 3114, 3312, 3548, 3753, 3953, 4251, 4386, 4731, 4802, 5073

Offset

1,3

Comments

Conjecture: This sequence is strictly increasing for n > 1.

Links

Peter Kagey and Giovanni Resta, Table of n, a(n) for n = 1..2000 (first 300 terms from Peter Kagey)

Example

Let p_n = (n, a(n)).
For n = 4, a(4) = 5 because
d(p_1, p_4) = 3 = d(p_1, p_3) if a(4) = 1,
d(p_3, p_4) = 1 = d(p_1, p_2) if a(4) = 2,
d(p_3, p_4) = 2 = d(p_2, p_3) if a(4) = 3,
d(p_3, p_4) = 3 = d(p_1, p_3) if a(4) = 4, therefore
a(4) = 5, the least value that does not create a contradiction.

Mathematica

dq[p_, q_] := Total@Abs[p - q]; good[w_] := Catch[Do[If[ MemberQ[di, dq[w, P[[i]]]], Throw@False], {i, Length@P}]; True]; di = P = {}; n = 0; While[n < 51, n++; k = 1; While[! good[{n, k}], k++]; di = Join[di, dq[{n, k}, #] & /@ P]; AppendTo[P, {n, k}]]; Last /@ P (* Giovanni Resta, Apr 06 2017 *)

Crossrefs

Cf. A005282 (Chebyshev distance), A284916 (Euclidean distance).

Sequence in context: A138226 A175287 A346822 * A007979 A097701 A362548

Adjacent sequences: A284914 A284915 A284916 * A284918 A284919 A284920

Keyword

nonn

Author

Peter Kagey, Apr 05 2017

Status

approved