A361644 - OEIS

%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