A333596 a(0) = 0; for n > 0, a(n) = a(n-1) + (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).
0, 1, 0, 1, 1, 3, 3, 5, 3, 3, 1, 1, 1, 3, 3, 5, 4, 5, 4, 5, 6, 9, 10, 13, 12, 13, 12, 13, 14, 17, 18, 21, 18, 17, 14, 13, 12, 13, 12, 13, 10, 9, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 9, 10, 13, 12, 13, 12, 13, 14, 17, 18, 21, 19, 19, 17, 17, 17, 19, 19, 21, 19, 19
Offset: 0
Examples
n in #odd #even n base 4 digits - digits + a(n-1) = a(n) = ====== =============================== 0 0 0 - 0 1 1 1 - 0 + 0 = 1 2 2 0 - 1 + 1 = 0 3 3 1 - 0 + 0 = 1 4 10 1 - 1 + 1 = 1 5 11 2 - 0 + 1 = 3 6 12 1 - 1 + 3 = 3 7 13 2 - 0 + 3 = 5
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 0, a(n-1) +add( `if`(i in [1, 3], 1, -1), i=convert(n, base, 4))) end: seq(a(n), n=0..80); # Alois P. Heinz, May 30 2020 -
Mathematica
f[n_] := Total[(-1)^(r = Range[0, 3]) * DigitCount[n, 4, r]]; a[0] = 0; a[n_] := a[n] = a[n - 1] - f[n]; Array[a, 100, 0] (* Amiram Eldar, Apr 24 2020 *)
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PARI
a(n) = my(v=digits(n+1,4),s=0); for(i=1,#v, my(t=v[i]); v[i]=t*s+!(t%2); s-=(-1)^t); fromdigits(v,4); \\ Kevin Ryde, May 30 2020
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PARI
b(n)=my(d=digits(n,4)); -sum(i=1,#d,(-1)^d[i]) first(n)=my(s); concat(0,vector(n,k,s+=b(k))) \\ Charles R Greathouse IV, Jul 04 2020
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Python
import numpy as np def qnary(n): e = n//4 q = n%4 if n == 0 : return 0 if e == 0 : return q if e != 0 : return np.append(qnary(e), q) m = 400 v = [0] for i in range(1,m+1) : t = np.array(qnary(i)) t[t%2 != 0] = 1 t[t%2 == 0] = -1 v = np.append(v, np.sum([np.sum(t), v[i-1]])) -
Python
from itertools import accumulate def A334841(n): return 2*bin(n)[-1:1:-2].count('1')-(len(bin(n))-1)//2 if n > 0 else 0 A333596_list = list(accumulate(A334841(n) for n in range(10000))) # Chai Wah Wu, Sep 03 2020
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R
qnary = function(n, e, q){ e = floor(n/4) q = n%%4 if(n == 0 ){return(0)} if(e == 0){return(q)} else{return(c(qnary(e), (q)))} } m = 400 s = seq(2,m) v = c(0) for(i in s){ x = qnary(i-1) x[which(x%%2!=0)] = 1 x[which(x%%2==0)] = -1 v[i] = sum(x,v[i-1]) }
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