A361645 -id:A361645 - cp's OEIS frontend

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

Rémy Sigrist, Mar 19 2023

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).

			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)

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

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

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); }

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

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

A361676 a(n) is the greatest k such that n appears in the k-th row of triangle A361644.

Original entry on oeis.org

0, 1, 2, 3, 5, 5, 6, 7, 11, 10, 10, 11, 13, 13, 14, 15, 23, 22, 21, 21, 21, 21, 22, 23, 27, 26, 26, 27, 29, 29, 30, 31, 47, 46, 45, 45, 43, 42, 42, 43, 43, 42, 42, 43, 45, 45, 46, 47, 55, 54, 53, 53, 53, 53, 54, 55, 59, 58, 58, 59, 61, 61, 62, 63, 95, 94, 93

Offset: 0

Rémy Sigrist, Mar 20 2023

All terms belong to A003754.
To compute a(n): consider the run lengths in the binary expansion of n (i.e. the n-th row of A101211) and replace from left to right each value v > 1 at even index with (1, v-1); at the end, there remain the run lengths in the binary expansion of a(n).
See A361645 for the least k's.

			The first terms, in decimal and in binary, are:
  n   a(n)  bin(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     1       1          1
   2     2      10         10
   3     3      11         11
   4     5     100        101
   5     5     101        101
   6     6     110        110
   7     7     111        111
   8    11    1000       1011
   9    10    1001       1010
  10    10    1010       1010
  11    11    1011       1011
  12    13    1100       1101
  13    13    1101       1101
  14    14    1110       1110
  15    15    1111       1111

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, PARI program
Index entries for sequences related to binary expansion of n

Cf. A000975, A003754, A101211, A361644, A361645.

PARI
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See Links section.
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a(n) >= n with equality iff n belongs to A003754.

a(n) >= A361645(n) with equality iff n belongs to A000975.

A361674 Irregular triangle T(n, k), n >= 0, k = 1..2^A092339(n), read by rows; the n-th row lists the numbers k such that n appears in the k-th row of A361644.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 5, 5, 6, 4, 5, 6, 7, 8, 9, 10, 11, 9, 10, 10, 10, 11, 10, 11, 12, 13, 10, 13, 9, 10, 13, 14, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 17, 18, 21, 22, 18, 21, 18, 19, 20, 21, 20, 21, 21, 21, 22, 20, 21, 22, 23, 20, 21, 22, 23, 24, 25, 26, 27

Offset: 0

Rémy Sigrist, Mar 20 2023

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 n are different then they are also different in k (i = 0 corresponding to the least significant bit).

			Triangle T(n, k) 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                   10  10
   3  2, 3                11  10, 11
   4  4, 5               100  100, 101
   5  5                  101  101
   6  5, 6               110  101, 110
   7  4, 5, 6, 7         111  100, 101, 110, 111
   8  8, 9, 10, 11      1000  1000, 1001, 1010, 1011
   9  9, 10             1001  1001, 1010
  10  10                1010  1010
  11  10, 11            1011  1010, 1011
  12  10, 11, 12, 13    1100  1010, 1011, 1100, 1101
  13  10, 13            1101  1010, 1101
  14  9, 10, 13, 14     1110  1001, 1010, 1101, 1110

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

Cf. A361644, A361645, A361676.

row(n) = { my (r = [n], m); for (e = 1, exponent(n), if (bittest(n, e-1)==bittest(n, e), m = 2^e-1; r = concat(r, [bitxor(v, m) | v <- r]););); vecsort(r); }

T(n, 1) = A361645(n).

T(n, 2^A092339(n)) = A361676(n).

cp's OEIS Frontend

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.

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A361676 a(n) is the greatest k such that n appears in the k-th row of triangle A361644.

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A361674 Irregular triangle T(n, k), n >= 0, k = 1..2^A092339(n), read by rows; the n-th row lists the numbers k such that n appears in the k-th row of A361644.

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Programs

PARI

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