A343963 a(0) = 0, and for any n > 0, the binary expansion of n has n digits and starts with the binary expansion of n, say of w digits, and in case n > w, the remaining binary digits in a(n) are those of a(n-w).

0, 1, 2, 7, 9, 22, 55, 121, 137, 310, 695, 1529, 3209, 6966, 15031, 32249, 34297, 72841, 154422, 326327, 687609, 1410553, 2956425, 6183734, 12909239, 26902009, 55936505, 116202633, 241064758, 499448503, 1033534969, 2136311289, 2203420153, 4545387657

Offset: 0

Author

Rémy Sigrist, May 05 2021

Keywords

• nonn
• base

Comments

To build the binary expansion of a(n):

- start with n indeterminate digits,

- while there are some, say m, indeterminate digits,

replace the first of them with the binary expansion of m.

The binary plot of the sequence has locally periodic patterns.

Examples

			For n = 10:
- the binary expansion of a(10) has 10 digits, and is the concatenation of:
   - the binary expansion of 10 which is "1010",
   - the binary expansion of 10 - 4 = 6 which is "110",
   - the binary expansion of 10 - 4 - 3 = 3 which is "11",
   - the binary expansion of 10 - 4 - 3 - 2 = 1 which is "1",
   - as 10 = 4 + 3 + 2 + 1, we stop here,
- so the binary expansion of a(10) is "1010110111",
- and a(10) = 695.
		

Links

• Rémy Sigrist, Binary plot of the sequence for n <= 2^10

Crossrefs

Cf. A070939, A319678.

Programs

PARI

a(n) = { if (n==0, 0, my (k=n-#binary(n)); n*2^k+a(k)) }

Formula

A070939(a(n)) = n for any n > 0.