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%I #18 Dec 12 2023 14:18:14
%S 1,1,2,5,9,16,25,38,58,72,87,112,136,169,190,237,274,344,383,456,545,
%T 572,640,752,798,891,944,989,1131,1283,1365,1492,1540,1788,1862,1994,
%U 2218,2342,2472,2741,2885,3114,3312,3548,3753,3953,4251,4386,4731,4802,5073
%N 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))).
%C Conjecture: This sequence is strictly increasing for n > 1.
%H Peter Kagey and Giovanni Resta, <a href="/A284917/b284917.txt">Table of n, a(n) for n = 1..2000</a> (first 300 terms from Peter Kagey)
%e Let p_n = (n, a(n)).
%e For n = 4, a(4) = 5 because
%e d(p_1, p_4) = 3 = d(p_1, p_3) if a(4) = 1,
%e d(p_3, p_4) = 1 = d(p_1, p_2) if a(4) = 2,
%e d(p_3, p_4) = 2 = d(p_2, p_3) if a(4) = 3,
%e d(p_3, p_4) = 3 = d(p_1, p_3) if a(4) = 4, therefore
%e a(4) = 5, the least value that does not create a contradiction.
%t 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 *)
%Y Cf. A005282 (Chebyshev distance), A284916 (Euclidean distance).
%K nonn
%O 1,3
%A _Peter Kagey_, Apr 05 2017
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