A320916 Consider A010060 as a 2-adic number ...100110010110, then a(n) is its approximation up to 2^n.

0, 0, 2, 6, 6, 22, 22, 22, 150, 406, 406, 406, 2454, 2454, 10646, 27030, 27030, 92566, 92566, 92566, 616854, 616854, 2714006, 6908310, 6908310, 6908310, 40462742, 107571606, 107571606, 376007062, 376007062, 376007062, 2523490710, 6818458006, 6818458006, 6818458006

Offset

0,3

Comments

This is another interpretation of A010060 as a number, in a different way as considering it as a binary number.

Consider the g.f. of A010060. As a real-valued (or complex-valued) function it only converges for |x| < 1. In 2-adic field it only converges for |x|_2 < 1 as well, but here |x|_2 is a different metric. For a 2-adic number x, |x|_2 < 1 iff x is an even 2-adic integer.

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{i=0..n-1} A010060(i)*2^i (empty sum yields 0 for n = 0).

Example

a(1) =     0_2 =  0.
a(2) =    10_2 =  2.
a(3) =   110_2 =  6.
a(4) =  0110_2 =  6.
a(5) = 10110_2 = 22.
...

Mathematica

With[{nmax = 50}, Table[FromDigits[#[[-n;; ]], 2], {n, 0, nmax}] & [ThueMorse[Range[nmax, 0, -1]]]] (* or *)
A320916[n_] := FromDigits[ThueMorse[Range[n-1, 0, -1]], 2]; Array[A320916, 51, 0] (* Paolo Xausa, Oct 18 2024 *)

Prog

(PARI) a(n) = sum(i=0, n-1, 2^i*(hammingweight(i)%2))

Crossrefs

Cf. A010060, A122570, A019300 (bit reversal).

Sequence in context: A367767 A368001 A258702 * A119551 A100634 A242527

Adjacent sequences: A320913 A320914 A320915 * A320917 A320918 A320919

Keyword

nonn,easy

Author

Jianing Song, Oct 26 2018

Status

approved