OFFSET
0,1
COMMENTS
The number of 0's in the binary representation of a(n) is at most 2. - Chai Wah Wu, Mar 06 2026
LINKS
Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..200 from Ruud H.G. van Tol)
Chai Wah Wu, A characterization of OEIS A393797
PROG
(PARI) a(n)= for(i=1, oo, my(d=binary(i)); if(n == 2*d*-[-#d..-1]~ - binomial(#d+1, 2), return(i)))
(PARI) a(n)= n||return(4); my(k=(sqrtint(n*8)+1)\2, t=binomial(k+1, 2)-n); if(t%2, t+=k++; if(!(k%2), t+=k++)); my(m=t\2, a=2^k-1); if(!m, a, m<k, a-2^(m-1), a-2^(k-2)-2^(m-k)); \\ port of the Python code; Ruud H.G. van Tol, Mar 08 2026
(Python)
from math import isqrt, comb
def A393797(n):
if n == 0: return 4
k = isqrt(n<<3)+1>>1
t = comb(k+1, 2)-n
if t&1:
k += 1
t += k
if k&1^1:
k += 1
t += k
m, a = t>>1, (1<<k)-1
if m == 0:
return a
elif m < k:
return a-(1<<m-1)
else:
return a-(1<<k-2)-(1<<m-k) # Chai Wah Wu, Mar 06 2026
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Ruud H.G. van Tol, Feb 27 2026
STATUS
approved
