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a(n) = smallest positive number not already in the sequence such that all n(n+1)/2 numbers in the triangle of differences of the first n terms are distinct.
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%I #50 Oct 09 2019 20:55:57

%S 1,3,9,5,12,10,23,8,22,17,42,16,43,20,38,26,45,32,65,28,64,39,76,34,

%T 81,48,98,40,92,54,109,60,116,51,114,58,117,70,136,67,135,71,145,72,

%U 147,69,146,80,164,87,166,82,170,108,198,99

%N a(n) = smallest positive number not already in the sequence such that all n(n+1)/2 numbers in the triangle of differences of the first n terms are distinct.

%C Inspired by A327743.

%C From _Rémy Sigrist_, Sep 25 2019: (Begin)

%C The sequence is finite, with 56 terms.

%C Let b and c be the first and second differences of a, respectively, hence:

%C - b(55) = a(56) - a(55) = 99 - 198 = -99,

%C - b(56) = a(57) - a(56) = a(57) - 99,

%C - c(55) = b(56) - b(55) = a(57), a contradiction.

%C (End)

%C Since this definition leads to a finite sequence, it is natural to ask instead for the "Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, all the k(k+1)/2 numbers in the triangle of differences of the first k terms are distinct." This is A327460.

%C If only first differences are considered, one gets the classical Mian-Chowla sequence A005282. - _M. F. Hasler_, Oct 09 2019

%e Difference triangle of the first k=8 terms of the sequence:

%e 1, 3, 9, 5, 12, 10, 23, 8, ...

%e 2, 6, -4, 7, -2, 13, -15, ...

%e 4, -10, 11, -9, 15, -28, ...

%e -14, 21, -20, 24, -43, ...

%e 35, -41, 44, -67, ...

%e -76, 85, -111, ...

%e 161, -196, ...

%e -357, ...

%e All 8*9/2 = 36 numbers are distinct.

%Y Cf. A327743, A327460.

%Y For first differences see A327458; for the leading column of the difference triangle see A327459.

%Y Cf. A005282.

%K nonn,full,fini

%O 1,2

%A _N. J. A. Sloane_, Sep 24 2019, revised Sep 25 2019.