A320916 - OEIS

%I #54 Oct 18 2024 06:18:43

%S 0,0,2,6,6,22,22,22,150,406,406,406,2454,2454,10646,27030,27030,92566,

%T 92566,92566,616854,616854,2714006,6908310,6908310,6908310,40462742,

%U 107571606,107571606,376007062,376007062,376007062,2523490710,6818458006,6818458006,6818458006

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

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

%C 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.

%H Paolo Xausa, <a href="/A320916/b320916.txt">Table of n, a(n) for n = 0..1000</a>

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

%e a(1) =     0_2 =  0.

%e a(2) =    10_2 =  2.

%e a(3) =   110_2 =  6.

%e a(4) =  0110_2 =  6.

%e a(5) = 10110_2 = 22.

%e ...

%t With[{nmax = 50}, Table[FromDigits[#[[-n;;]], 2], {n, 0, nmax}] & [ThueMorse[Range[nmax, 0, -1]]]] (* or *)

%t A320916[n_] := FromDigits[ThueMorse[Range[n-1, 0, -1]], 2]; Array[A320916, 51, 0] (* _Paolo Xausa_, Oct 18 2024 *)

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

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

%K nonn,easy

%O 0,3

%A _Jianing Song_, Oct 26 2018