A361644 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.

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, 11, 12, 15, 12, 15, 12, 13, 14, 15, 14, 15, 15, 16, 31, 16, 17, 30, 31, 16, 17, 18, 19, 28, 29, 30, 31, 16, 19, 28, 31, 16, 19, 20, 23, 24, 27, 28, 31

Offset

0,3

Comments

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).

The value m appears 2^A092339(m) times in the triangle (see A361674).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..9841 (rows for n = 0..511 flattened)

Index entries for sequences related to binary expansion of n

Formula

T(n, 1) = A342126(n).

T(n, max(1, 2^(A005811(n)-1))) = A003817(n).

Example

Triangle begins (in decimal and in binary):
  n   n-th row      bin(n)  n-th row in binary
  --  ------------  ------  ------------------
   0  0                  0  0
   1  1                  1  1
   2  2, 3              10  10, 11
   3  3                 11  11
   4  4, 7             100  100, 111
   5  4, 5, 6, 7       101  100, 101, 110, 111
   6  6, 7             110  110, 111
   7  7                111  111
   8  8, 15           1000  1000, 1111
   9  8, 9, 14, 15    1001  1000, 1001, 1110, 1111
.
For n = 9:
- the binary expansion of 9 is "1001",
- the corresponding run lengths are 1, 2, 1,
- so the 9th row contains the values with the following run lengths:
      1, 2, 1  ->   9 ("1001" in binary)
      1,  2+1  ->   8 ("1000" in binary)
      1+2,  1  ->  14 ("1110" in binary)
       1+2+1   ->  15 ("1111" in binary)

Prog

(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); }

Crossrefs

Cf. A003817, A005811, A092339, A101211, A225081, A342126, A361645, A361646, A361674, A361676.

Sequence in context: A132289 A078467 A154217 * A185738 A239361 A266362

Adjacent sequences: A361641 A361642 A361643 * A361645 A361646 A361647

Keyword

nonn,base,tabf

Author

Rémy Sigrist, Mar 19 2023

Status

approved