A269735 G.f.: Sum_{k >= 0} x^(2^k)*(1-x^(2^k))/(1+x^(2^k)).

0, 1, -1, 2, -3, 2, 0, 2, -5, 2, 0, 2, -2, 2, 0, 2, -7, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 2, -2, 2, 0, 2, -9, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 2, -2, 2, 0, 2, -6, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 2, -2, 2, 0, 2, -11, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 2, -2, 2, 0, 2, -6, 2, 0, 2, -2, 2, 0, 2, -4, 2, 0, 2, -2

Offset: 0

Author

N. J. A. Sloane, Mar 11 2016

Keywords

• sign

Comments

Second differences of A268289.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Cf. A268289, A187038.

Programs

Maple

t7:=add(x^(2^k)*(1-x^(2^k))/(1+x^(2^k)),k=0..12);
t8:=series(t7,x,256);
# second Maple program:
b:= proc(n) option remember; `if`(n<0, 0,
      add(2*i-1, i=Bits[Split](n)))
    end:
a:= n-> b(n)-b(n-1):
seq(a(n), n=0..92);  # Alois P. Heinz, Jan 18 2022

Mathematica

Join[{0, 0}, Table[DigitCount[n, 2, 1] - DigitCount[n, 2, 0], {n, 1, 100}]] // Differences (* Jean-François Alcover, Jun 27 2022 *)

PARI

up_to = 1024;
A268289list(up_to) =  { my(v=vector(up_to), s = 1); v[1] = s; for(n=2, up_to, s += (2*hammingweight(n) - #binary(n)); v[n] = s); (v); };
v268289 = A268289list(up_to+1);
A268289(n) = if(!n,n,v268289[n]);
almost_firstdiffs_of_A268289(n) = if(!n,1,v268289[n+1]-v268289[n]);
A269735(n) = if(n<=1,n,almost_firstdiffs_of_A268289(n-1)-almost_firstdiffs_of_A268289(n-2)); \\ Antti Karttunen, Sep 30 2018

PARI

up_to_k = 16;
up_to = 1+(2^up_to_k);
x='x+O('x^(up_to+1));
v269735 = Vec(sum(k=0,up_to_k,x^(2^k)*(1-x^(2^k))/(1+x^(2^k))));
A269735(n) = if(!n,n,v269735[n]); \\ Antti Karttunen, Oct 01 2018