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A384002 Let S(n,j,k), j = 1..n, k = 1..A024718(n), where row 1 = {(0),(1)}, and row n = union of n-tuples whose sum m < n, and the n-tuples formed by appending m to the (n-1)-tuples in row n-1. Then T(n,j) = j-th tuple in row n of S read as a base n+1 number expressed in decimal.

%I A384002 #9 Jul 16 2025 20:08:33
%S A384002 0,1,0,1,2,3,4,0,1,2,3,4,5,6,8,9,16,17,18,20,21,32,0,1,2,3,4,5,6,7,8,
%T A384002 10,11,12,15,16,25,26,27,28,30,31,32,35,36,50,51,52,55,56,75,125,126,
%U A384002 127,128,130,131,132,135,136,150,151,152,155,156,175,250,251,252,255,275,375
%N A384002 Let S(n,j,k), j = 1..n, k = 1..A024718(n), where row 1 = {(0),(1)}, and row n = union of n-tuples whose sum m < n, and the n-tuples formed by appending m to the (n-1)-tuples in row n-1. Then T(n,j) = j-th tuple in row n of S read as a base n+1 number expressed in decimal.
%H A384002 Michael De Vlieger, <a href="/A384002/b384002.txt">Table of n, a(n) for n = 1..12029</a> (rows n = 1..8, flattened).
%H A384002 Michael De Vlieger, <a href="/A384002/a384002.png">Log log scatterplot of a(n)</a> for n = 1..2479069 (rows n = 1..12), ignoring a(n) = 0.
%F A384002 T(n,j) = base n+1 expansion of j-th tuple in row n of A384001.
%F A384002 Length of row n = A024718(n).
%e A384002 Table begins:
%e A384002   1:  0, 1;
%e A384002   2:  0, 1, 2, 3, 4;
%e A384002   3:  0, 1, 2, 3, 4, 5, 6, 8, 9, 16, 17, 18, 20, 21, 32;
%e A384002   4:  0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15, 16, 25, 26, 27, 28, 30, 31, 32,
%e A384002       35, 36, 50, 51, 52, 55, 56, 75, 125, 126, 127, 128, 130, 131, 132, 135,
%e A384002       136, 150, 151, 152, 155, 156, 175, 250, 251, 252, 255, 275, 375;
%e A384002   etc.
%e A384002 Row 2 of S is {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)}. Reading the tuples in row 2 as a base 3 number, we have row 2 of this sequence.
%t A384002 nn = 8; w[0] = {{0}};
%t A384002 Do[If[n == 1, Set[w[1], {{0}, {1}}],
%t A384002   Set[w[n], Union@ Join[Select[Tuples[Range[0, n - 1], n], Total[#] < n &],
%t A384002     Map[Append[#, n - Total[#]] &, w[n - 1] ] ] ] ], {n, nn}];
%t A384002 Table[Map[FromDigits[#, n + 1] &, w[n]], {n, 0, nn}]
%Y A384002 Cf. A024718, A384001.
%K A384002 nonn,tabf,base,easy,look
%O A384002 1,5
%A A384002 _Michael De Vlieger_, May 21 2025