A355653 For any number n with runs in binary expansion (r_w, ..., r_0), let p(n) be the polynomial of a single indeterminate x where the coefficient of x^e is r_e for e = 0..w and otherwise 0, and let q be the inverse of p; a(n) = q(p(n)').

0, 0, 1, 0, 1, 6, 3, 0, 1, 12, 57, 6, 3, 30, 7, 0, 1, 24, 225, 12, 57, 966, 115, 6, 3, 60, 505, 30, 7, 126, 15, 0, 1, 48, 897, 24, 225, 7686, 451, 12, 57, 1932, 31801, 966, 115, 3870, 231, 6, 3, 120, 2017, 60, 505, 16326, 1011, 30, 7, 252, 4089, 126, 15, 510

Offset: 0

Author

Rémy Sigrist, Jul 12 2022

Keywords

• nonn
• base

Comments

In other words, a(n) encodes the derivative of the polynomial encoded by n.

For n > 0, the runs in binary expansion of a(n) are (w*r_w, ..., 1*r_1).

Examples

			The first terms, alongside their binary expansions and corresponding polynomials, are:
  n   a(n)  bin(n)  bin(a(n))  p(n)               p(a(n))
  --  ----  ------  ---------  -----------------  ---------------
   0     0       0          0                  0                0
   1     0       1          0                  1                0
   2     1      10          1              x + 1                1
   3     0      11          0                  2                0
   4     1     100          1              x + 2                1
   5     6     101        110        x^2 + x + 1          2*x + 1
   6     3     110         11            2*x + 1                2
   7     0     111          0                  3                0
   8     1    1000          1              x + 3                1
   9    12    1001       1100      x^2 + 2*x + 1          2*x + 2
  10    57    1010     111001  x^3 + x^2 + x + 1  3*x^2 + 2*x + 1
  11     6    1011        110        x^2 + x + 2          2*x + 1
  12     3    1100         11            2*x + 2                2
  13    30    1101      11110      2*x^2 + x + 1          4*x + 1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192
• Rémy Sigrist, Binary plot of the sequence for n = 0..2^10
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000225, A005811, A101211, A355654.

Programs

PARI

toruns(n) = { my (r=[]); while (n, my (v=valuation(n+n%2, 2)); n\=2^v; r=concat(v, r)); r }
fromruns(r) = { my (v=0); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }
a(n) = { fromruns(Vec(deriv(Pol(toruns(n))))) }

Formula

a^A005811(n)(n) = 0 (where a^k denotes the k-th iterate of a).

a(n) = 0 iff n belongs to A000225.

a(n) = 1 iff n = 2^k for some k > 0.

A005811(a(n)) = A005811(n)-1 for any n > 0.