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A334841 a(0) = 0; for n > 0, a(n) = (number of 1's and 3's in base 4 representation of n) - (number of 0's and 2's in base 4 representation of n).

%I A334841 #35 Jan 27 2025 08:47:41
%S A334841 0,1,-1,1,0,2,0,2,-2,0,-2,0,0,2,0,2,-1,1,-1,1,1,3,1,3,-1,1,-1,1,1,3,1,
%T A334841 3,-3,-1,-3,-1,-1,1,-1,1,-3,-1,-3,-1,-1,1,-1,1,-1,1,-1,1,1,3,1,3,-1,1,
%U A334841 -1,1,1,3,1,3,-2,0,-2,0,0,2,0,2,-2,0,-2,0,0,2,0,2,0,2,0,2,2,4,2,4,0
%N A334841 a(0) = 0; for n > 0, a(n) = (number of 1's and 3's in base 4 representation of n) - (number of 0's and 2's in base 4 representation of n).
%C A334841 Values are even for base 4 representations of n with an even number of digits, and odd for base 4 representations of n with an odd number of digits, except for a(0).
%F A334841 a(n) = 2*A139351(n) - A110591(n), n>0. - _R. J. Mathar_, Sep 02 2020
%e A334841       n in    #odd    #even
%e A334841   n  base 4  digits - digits = a(n)
%e A334841   =  ======  =======================
%e A334841   0    0        0   -            0
%e A334841   1    1        1   -    0   =   1
%e A334841   2    2        0   -    1   =  -1
%e A334841   3    3        1   -    0   =   1
%e A334841   4   10        1   -    1   =   0
%e A334841   5   11        2   -    0   =   2
%e A334841   6   12        1   -    1   =   0
%e A334841   7   13        2   -    0   =   2
%p A334841 a:= n-> `if`(n=0, 0, add(`if`(i in [1, 3], 1, -1), i=convert(n, base, 4))):
%p A334841 seq(a(n), n=0..100);  # _Alois P. Heinz_, May 30 2020
%t A334841 a[0] = 0; a[n_] := Total[-(-1)^(r = Range[0, 3]) * DigitCount[n, 4, r]]; Array[a, 100, 0] (* _Amiram Eldar_, May 13 2020 *)
%t A334841 Join[{0},Table[Total[If[EvenQ[#],-1,1]&/@IntegerDigits[n,4]],{n,90}]] (* _Harvey P. Dale_, Sep 06 2020 *)
%o A334841 (R)
%o A334841 qnary = function(n, e, q){
%o A334841   e = floor(n/4)
%o A334841   q = n%%4
%o A334841   if(n == 0 ){return(0)}
%o A334841   if(e == 0){return(q)}
%o A334841   else{return(c(qnary(e), (q)))}
%o A334841 }
%o A334841 m = 400
%o A334841 s = seq(2, m)
%o A334841 v = c(0)
%o A334841 for(i in s){
%o A334841   x = qnary(i-1)
%o A334841   x[which(x%%2!=0)] = 1
%o A334841   x[which(x%%2==0)] = -1
%o A334841   v[i] = sum(x)
%o A334841 }
%o A334841 (Python)
%o A334841 import numpy as np
%o A334841 def qnary(n):
%o A334841     e = n//4
%o A334841     q = n%4
%o A334841     if n == 0 : return 0
%o A334841     if e == 0 : return q
%o A334841     if e != 0 : return np.append(qnary(e), q)
%o A334841 m = 400
%o A334841 v = [0]
%o A334841 for i in range(1, m+1) :
%o A334841     t = np.array(qnary(i))
%o A334841     t[t%2 != 0] = 1
%o A334841     t[t%2 == 0] = -1
%o A334841     v = np.append(v, np.sum(t))
%o A334841 (PARI) a(n) = my(ret=0); if(n,forstep(i=0,logint(n,2),2, if(bittest(n,i),ret++,ret--))); ret; \\ _Kevin Ryde_, May 24 2020
%o A334841 (Python)
%o A334841 def A334841(n):
%o A334841     return 2*bin(n)[-1:1:-2].count('1')-(len(bin(n))-1)//2 if n > 0 else 0 # _Chai Wah Wu_, Sep 03 2020
%Y A334841 Cf. A053737, A010065, A037863, A007090, A301336, A333596.
%K A334841 sign,easy,base
%O A334841 0,6
%A A334841 _Alexander Van Plantinga_, May 13 2020