A318926 Take the binary expansion of n, starting with the least significant bit, and concatenate the lengths of the runs.

1, 11, 2, 21, 111, 12, 3, 31, 121, 1111, 211, 22, 112, 13, 4, 41, 131, 1121, 221, 2111, 11111, 1211, 311, 32, 122, 1112, 212, 23, 113, 14, 5, 51, 141, 1131, 231, 2121, 11121, 1221, 321, 3111, 12111, 111111, 21111, 2211, 11211, 1311, 411, 42, 132, 1122, 222, 2112, 11112, 1212, 312, 33, 123

Offset

1,2

Comments

Obviously this compressed notation is useful only for n < 2047. A227736 is a version which works for all n. [Corrected by M. F. Hasler, Mar 12 2025]

Links

Paolo Xausa, Table of n, a(n) for n = 1..10000

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - N. J. A. Sloane, Sep 09 2018. See Procedure 1.

Example

n, binary, run lengths, -> a(n)
1, [1], [1] -> 1
2, [0, 1], [1, 1] -> 11
3, [1, 1], [2] ->  2
4, [0, 0, 1], [2, 1] -> 21
5, [1, 0, 1], [1, 1, 1] -> 111
6, [0, 1, 1], [1, 2] -> 12
7, [1, 1, 1], [3] -> 3
8, [0, 0, 0, 1], [3, 1] ->  31,
...
From M. F. Hasler, Mar 12 2025: (Start)
For n = 1023 = 2^10-1, n = '1'*10 in binary, so there is only one run of length 10, whence a(n) = 10. This value cannot occur at any other index n.
For n = 1024 = 2^10, n = '1'+'0'*10 in binary, so the run lengths, from right to left, are [10, 1], whence a(n) = 101. The only other index n for which this value occurs is n = 2^101-1.
For n = 1025 = 2^10+1, n = '1'+'0'*9+'1' in binary, so a(n) = 191. This values occurs for the second time as a(n = 2^19), for the third time for a(n = 2^92-2), and for the 4th and last time as a(n = 2^191-1).
Similarly, a(1026) = 1181 appears for the second time at n = 2^19 + 1 = 524289;
  a(1027) = 281 occurs a 2nd, 3rd and 4th time at n = 2^28, (2^81-1)*2 and 2^281-1.
The first duplicate value occurs as a(2047 = 2^11-1) = 11 = a(2). (End)

Mathematica

A318926[n_] := FromDigits[Flatten[IntegerDigits[Map[Length, Split[Reverse[IntegerDigits[n, 2]]]]]]];
Array[A318926, 100] (* Paolo Xausa, Mar 16 2025 *)

Prog

(Python)
from itertools import groupby
def A318926(n): return int(''.join(str(len(list(g))) for k, g in groupby(bin(n)[:1:-1]))) # Chai Wah Wu, Mar 11 2022
(PARI) A318926(n)=eval(strjoin(Vecrev(A101211_row(n)))); \\ M. F. Hasler, Mar 11 2025

Crossrefs

Cf. A227736 (run lengths in rows instead of concatenation), A101211 (rows in reverse order), A318927 (concatenation in reverse order).

Sequence in context: A262369 A092260 A370004 * A336874 A040120 A365220

Adjacent sequences: A318923 A318924 A318925 * A318927 A318928 A318929

Keyword

nonn,base,changed

Author

N. J. A. Sloane, Sep 09 2018

Extensions

More terms from M. F. Hasler, Mar 12 2025

Status

approved