OFFSET
1,1
COMMENTS
Also numbers of the form (4^a)*b - 1 with positive integer a and odd integer b. The sequence has linear growth and the limit of a(n)/n is 6. - Stefan Steinerberger, Dec 18 2007
Evil and odious terms alternate. - Vladimir Shevelev, Jun 22 2009
Also odd numbers of the form m = (A079523(k)-1)/2. - Vladimir Shevelev, Jul 06 2009
As a set, this is the complement of A079523 in the odd numbers. - Michel Dekking, Feb 13 2019
From Ctibor O. Zizka, Dec 28 2024: (Start)
Numbers k >= 1 such that (k + 1)*(k + 2*r)/2 is not a square for any r >= 1.
Numbers k such that A076114(k + 1) = 0. (End)
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Thomas Zaslavsky, Anti-Fibonacci Numbers: A Formula, Sep 26 2016.
FORMULA
a(n) = 2*A079523(n) + 1. - Michel Dekking, Feb 13 2019
EXAMPLE
11 in binary is 1011, which ends with two 1's.
MAPLE
N:= 1000: # to get all terms up to N
Odds:= [seq(2*i+1, i=0..floor((N-1)/2)]:
f:= proc(n) local L, x;
L:= convert(n, base, 2);
x:= ListTools:-Search(0, L);
if x = 0 then type(nops(L), even) else type(x, odd) fi
end proc:
A131323:= select(f, Odds); # Robert Israel, Apr 02 2014
MATHEMATICA
Select[Range[500], OddQ[ # ] && EvenQ[FactorInteger[ # + 1][[1, 2]]] &] (* Stefan Steinerberger, Dec 18 2007 *)
en1Q[n_]:=Module[{ll=Last[Split[IntegerDigits[n, 2]]]}, Union[ll] =={1} &&EvenQ[Length[ll]]]; Select[Range[1, 501, 2], en1Q] (* Harvey P. Dale, May 18 2011 *)
PROG
(PARI) is(n)=n%2 && valuation(n+1, 2)%2==0 \\ Charles R Greathouse IV, Aug 20 2013
(Python)
from itertools import count, islice
def A131323_gen(startvalue=3): # generator of terms >= startvalue
return map(lambda n:(n<<1)+1, filter(lambda n:(~(n+1)&n).bit_length()&1, count(max(startvalue>>1, 1))))
(Python)
def A131323(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x):
c, s = n+x, bin(x+1)[2:]
l = len(s)
for i in range(l&1, l, 2):
c -= int(s[i])+int('0'+s[:i], 2)
return c
return bisection(f, n, n)<<1|1 # Chai Wah Wu, Jan 29 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Nadia Heninger and N. J. A. Sloane, Dec 16 2007
EXTENSIONS
More terms from Stefan Steinerberger, Dec 18 2007
STATUS
approved