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A350605 Irregular triangle read by rows: row n lists the elements of the set S_n in increasing order, where S_1 = {1}, and S_{n+1} is the set {k, 2*k+1, 3*k+1: k in S_n}.

%I A350605 #20 May 06 2022 19:53:58
%S A350605 1,1,3,4,1,3,4,7,9,10,13,1,3,4,7,9,10,13,15,19,21,22,27,28,31,40,1,3,
%T A350605 4,7,9,10,13,15,19,21,22,27,28,31,39,40,43,45,46,55,57,58,63,64,67,81,
%U A350605 82,85,94,121,1,3,4,7,9,10,13,15,19,21,22,27,28,31,39,40,43,45,46,55,57,58,63,64,67,79,81,82,85,87,91,93,94,111,115,117,118,121,127,129,130,135,136,139,163,165,166,171,172,175,189,190,193,202,243,244,247,256,283,364
%N A350605 Irregular triangle read by rows: row n lists the elements of the set S_n in increasing order, where S_1 = {1}, and S_{n+1} is the set {k, 2*k+1, 3*k+1: k in S_n}.
%C A350605 Row n has A350606(n) elements.
%C A350605 The rows converge to A002977.
%H A350605 Alois P. Heinz, <a href="/A350605/b350605.txt">Rows n = 1..15, flattened</a>
%H A350605 David A. Klarner and Richard Rado, <a href="https://www.jstor.org/stable/2318772">Linear combinations of sets of consecutive integers</a>, The American Mathematical Monthly, Vol. 80, No. 9 (1973), pp. 985-989.
%e A350605 The first few sets S_n are:
%e A350605 [1],
%e A350605 [1, 3, 4],
%e A350605 [1, 3, 4, 7, 9, 10, 13],
%e A350605 [1, 3, 4, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 40],
%e A350605 [1, 3, 4, 7, 9, 10, 13, 15, 19, 21, 22, 27, 28, 31, 39, 40, 43, 45, 46, 55, 57, 58, 63, 64, 67, 81, 82, 85, 94, 121],
%e A350605 ...
%p A350605 T:= proc(n) option remember; `if`(n=1, 1, sort(
%p A350605      [map(k-> [k, 2*k+1, 3*k+1][], {T(n-1)})[]])[])
%p A350605     end:
%p A350605 seq(T(n), n=1..6);  # _Alois P. Heinz_, Jan 12 2022
%t A350605 T[n_] := T[n] = If[n==1, {1}, {#, 2#+1, 3#+1}& /@ T[n-1] // Flatten //
%t A350605      Union];
%t A350605 Table[T[n], {n, 1, 6}] // Flatten (* _Jean-François Alcover_, May 06 2022, after _Alois P. Heinz_ *)
%o A350605 (Python)
%o A350605 from itertools import chain, islice
%o A350605 def A350605_gen(): # generator of terms
%o A350605     s = {1}
%o A350605     while True:
%o A350605         yield from sorted(s)
%o A350605         s = set(chain.from_iterable((x,2*x+1,3*x+1) for x in s))
%o A350605 A350605_list = list(islice(A350605_gen(),30)) # _Chai Wah Wu_, Jan 12 2022
%Y A350605 Cf. A002977, A350604, A350606.
%K A350605 nonn,look,tabf
%O A350605 1,3
%A A350605 _N. J. A. Sloane_, Jan 12 2022