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A339674 Irregular triangle T(n, k), n, k >= 0, read by rows; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; row n corresponds to the numbers k such that R(k) is included in R(n), in ascending order.

%I A339674 #23 Feb 24 2021 08:20:22
%S A339674 0,0,1,0,1,2,3,0,3,0,3,4,7,0,1,2,3,4,5,6,7,0,1,6,7,0,7,0,7,8,15,0,1,6,
%T A339674 7,8,9,14,15,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,0,3,4,7,8,11,12,15,
%U A339674 0,3,12,15,0,1,2,3,12,13,14,15,0,1,14,15,0
%N A339674 Irregular triangle T(n, k), n, k >= 0, read by rows; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; row n corresponds to the numbers k such that R(k) is included in R(n), in ascending order.
%C A339674 For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
%C A339674 The underlying idea is to take some or all of the rightmost runs of a number, and possibly merge some of them.
%C A339674 For any n >= 0, the n-th row:
%C A339674 - has 2^A000120(A003188(n)) terms,
%C A339674 - has first term 0 and last term A003817(n),
%C A339674 - has n at position A090079(n),
%C A339674 - corresponds to the distinct terms in n-th row of table A341840.
%H A339674 Rémy Sigrist, <a href="/A339674/b339674.txt">Table of n, a(n) for n = 0..6560</a>
%H A339674 Rémy Sigrist, <a href="/A339674/a339674.png">Scatterplot of (n, T(n, k)) for n <= 2^10</a>
%H A339674 Rémy Sigrist, <a href="/A339674/a339674.gp.txt">PARI program for A339674</a>
%H A339674 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A339674 T(n, 0) = 0.
%F A339674 T(n, A090079(n)) = n.
%F A339674 T(n, 2^A000120(A003188(n))-1) = A003817(n).
%e A339674 The triangle starts:
%e A339674     0;
%e A339674     0, 1;
%e A339674     0, 1, 2, 3;
%e A339674     0, 3;
%e A339674     0, 3, 4, 7;
%e A339674     0, 1, 2, 3, 4, 5, 6, 7;
%e A339674     0, 1, 6, 7;
%e A339674     0, 7;
%e A339674     0, 7, 8, 15;
%e A339674     0, 1, 6, 7, 8, 9, 14, 15;
%e A339674     0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;
%e A339674     0, 3, 4, 7, 8, 11, 12, 15;
%e A339674     0, 3, 12, 15;
%e A339674     0, 1, 2, 3, 12, 13, 14, 15;
%e A339674     0, 1, 14, 15;
%e A339674     0, 15;
%e A339674     ...
%o A339674 (PARI) See Links section.
%Y A339674 Cf. A000120, A003188, A003817, A090079, A227736, A341840.
%K A339674 nonn,base,tabf
%O A339674 0,6
%A A339674 _Rémy Sigrist_, Feb 21 2021