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%I #17 Mar 21 2023 15:03:56
%S 0,1,2,3,3,4,7,4,5,6,7,6,7,7,8,15,8,9,14,15,8,9,10,11,12,13,14,15,8,
%T 11,12,15,12,15,12,13,14,15,14,15,15,16,31,16,17,30,31,16,17,18,19,28,
%U 29,30,31,16,19,28,31,16,19,20,23,24,27,28,31
%N Irregular triangle T(n, k), n >= 0, k = 1..max(1, 2^(A005811(n)-1)), read by rows; the n-th row lists the integers with the same binary length as n and whose partial sums of run lengths are included in those of n.
%C In other words, the n-th row contains the numbers k with the same binary length as n and for any i >= 0, if the i-th bit and the (i+1)-th bit in k are different then they are also different in n (i = 0 corresponding to the least significant bit).
%C The value m appears 2^A092339(m) times in the triangle (see A361674).
%H Rémy Sigrist, <a href="/A361644/b361644.txt">Table of n, a(n) for n = 0..9841</a> (rows for n = 0..511 flattened)
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F T(n, 1) = A342126(n).
%F T(n, max(1, 2^(A005811(n)-1))) = A003817(n).
%e Triangle begins (in decimal and in binary):
%e n n-th row bin(n) n-th row in binary
%e -- ------------ ------ ------------------
%e 0 0 0 0
%e 1 1 1 1
%e 2 2, 3 10 10, 11
%e 3 3 11 11
%e 4 4, 7 100 100, 111
%e 5 4, 5, 6, 7 101 100, 101, 110, 111
%e 6 6, 7 110 110, 111
%e 7 7 111 111
%e 8 8, 15 1000 1000, 1111
%e 9 8, 9, 14, 15 1001 1000, 1001, 1110, 1111
%e .
%e For n = 9:
%e - the binary expansion of 9 is "1001",
%e - the corresponding run lengths are 1, 2, 1,
%e - so the 9th row contains the values with the following run lengths:
%e 1, 2, 1 -> 9 ("1001" in binary)
%e 1, 2+1 -> 8 ("1000" in binary)
%e 1+2, 1 -> 14 ("1110" in binary)
%e 1+2+1 -> 15 ("1111" in binary)
%o (PARI) row(n) = { my (r = []); while (n, my (v = valuation(n+n%2, 2)); n \= 2^v; r = concat(v, r)); my (s = [if (#r, 2^r[1]-1, 0)]); for (k = 2, #r, s = concat(s * 2^r[k], [(h+1)*2^r[k]-1|h<-s]);); vecsort(s); }
%Y Cf. A003817, A005811, A092339, A101211, A225081, A342126, A361645, A361646, A361674, A361676.
%K nonn,base,tabf
%O 0,3
%A _Rémy Sigrist_, Mar 19 2023
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