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A327743 a(n) = smallest positive number not already in the sequence such that for each k = 1, ..., n-1, the k-th differences are distinct. 9

%I #49 Oct 26 2019 04:41:34

%S 1,2,4,3,6,11,5,9,7,13,10,18,8,15,27,14,23,12,22,17,28,16,29,20,34,19,

%T 35,21,36,32,24,42,26,43,25,44,66,33,53,30,51,31,54,37,61,39,64,38,67,

%U 40,70,41,68,47,75,50,76,45,77,49,80,48,81,46,82,52,86

%N a(n) = smallest positive number not already in the sequence such that for each k = 1, ..., n-1, the k-th differences are distinct.

%C Is this sequence a permutation of the positive integers?

%C Does each k-th difference contain all nonzero integers?

%C It is not difficult to show that if a(1), ..., a(k) satisfy the requirements, then any sufficiently large number is a candidate for a(k+1). So a(k) exists for all k. - _N. J. A. Sloane_, Sep 24 2019

%C The original definition was "Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, the k-th differences are distinct."

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

%H Peter Kagey, <a href="/A327743/b327743.txt">Table of n, a(n) for n = 1..5000</a>

%e Illustration of the first eight terms of the sequence.

%e k | k-th differences

%e --+---------------------------------

%e 0 | 1, 2, 4, 3, 6, 11, 5, 9

%e 1 | 1, 2, -1, 3, 5, -6, 4

%e 2 | 1, -3, 4, 2, -11, 10

%e 3 | -4, 7, -2, -13, 21

%e 4 | 11, -9, -11, 34

%e 5 | -20, -2, 45

%e 6 | 18, 47

%e 7 | 29

%t a[1] = 1;

%t a[n_] := a[n] = For[aa = Array[a, n-1]; an = 1, True, an++, If[FreeQ[aa, an], aa = Append[aa, an]; If[AllTrue[Range[n-1], Unequal @@ Differences[ aa, #]&], Return[an]]]];

%t a /@ Range[1, 100] (* _Jean-François Alcover_, Oct 26 2019 *)

%Y Cf. A175498.

%Y First differences: A327452; leading column of difference triangle: A327457.

%Y If ALL terms of the difference triangle must be distinct, see A327460 and A327762.

%Y Cf. A005282.

%K nonn,nice

%O 1,2

%A _Peter Kagey_, Sep 24 2019

%E "Infinite" added to definition (for otherwise the one-term sequence 1 is earlier). - _N. J. A. Sloane_, Sep 25 2019

%E Changed definition to avoid use of "Lexicographically earliest infinite sequence" and the associated existence questions. - _N. J. A. Sloane_, Sep 28 2019

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)