OFFSET
1,1
COMMENTS
Either of form 2*a(m) or 4k+3, k >= 0, 0 < m < n.
a(n) such that A038189(a(n)) = 1.
Numbers n such that Kronecker(-n, m) = Kronecker(m, n) for all m. - Michael Somos, Sep 22 2005
From Antti Karttunen, Feb 20-21 2015: (Start)
Gives all n for which A005811(n) - A005811(n-1) = -1, from which follows that a(n) = the least k such that A255070(k) = n.
Gives the positions of even terms in A003602. (End)
Indices of negative terms in A164677. - M. F. Hasler, Aug 06 2015
Indices of the 0's in A014577. - Gabriele Fici, Jun 02 2016
Also indices of -1 in A034947. - Jianing Song, Apr 24 2021
Conjecture: alternate definition of same sequence is that a(1)=3 and a(n) is the smallest number > a(n-1) so that no number that is the sum of at most 2 terms in this sequence is a power of 2. - J. Lowell, Jan 20 2024
The asymptotic density of this sequence is 1/2. - Amiram Eldar, Aug 31 2024
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Allouche and J. Shallit, On three conjectures of P. Barry, arxiv preprint arXiv:2006.04708 [math.NT], June 8 2020.
Paul Barry, Some observations on the Rueppel sequence and associated Hankel determinants, arXiv:2005.04066 [math.CO], 2020.
Kevin Ryde, Iterations of the Dragon Curve, see index TurnRight, with a(n) = TurnRight(n-1).
FORMULA
Other identities. For all n >= 1 it holds that:
A014707(a(n) + 1) = 1. - Reinhard Zumkeller, Sep 28 2011
A055975(a(n)) < 0. - Reinhard Zumkeller, Apr 28 2012
From Antti Karttunen, Feb 20-21 2015: (Start)
a(n) = A246590(n)/2.
a(n) = 1 + A255068(n-1). (End)
MATHEMATICA
Select[Range[150], Mod[# / 2^IntegerExponent[#, 2], 4] == 3 &] (* Amiram Eldar, Aug 31 2024 *)
PROG
(PARI) for(n=1, 200, if(((n/2^valuation(n, 2)-1)/2)%2, print1(n", ")))
(PARI) {a(n) = local(m, c); if( n<1, 0, c=0; m=1; while( c<n, m++; if( ((m/2^valuation(m, 2) - 1) / 2) % 2, c++)); m)}; /* Michael Somos, Sep 22 2005 */
(PARI) is_A091067(n)=bittest(n, valuation(n, 2)+1) \\ M. F. Hasler, Aug 06 2015
(PARI) a(n) = my(t=1); n<<=1; forstep(i=logint(n, 2), 0, -1, if(bittest(n, i)==t, n++; t=!t)); n; \\ Kevin Ryde, Mar 21 2021
(Haskell)
import Data.List (elemIndices)
a091067 n = a091067_list !! (n-1)
a091067_list = map (+ 1) $ elemIndices 1 a014707_list
-- Reinhard Zumkeller, Sep 28 2011
(Scheme, with Antti Karttunen's IntSeq-library, two versions)
;; Antti Karttunen, Feb 20 2015
CROSSREFS
Essentially one less than A060833.
Characteristic function: A038189.
Complement of A091072.
First differences are in A106836 (from its second term onward).
Sequence A246590 gives the even terms.
Gives the positions of records (after zero) for A255070 (equally, the position of the first n there).
Cf. A106837 (gives n such that both n and n+1 are terms of this sequence).
Cf. A098502 (gives n such that both n and n+2 are, but n+1 is not in this sequence).
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Feb 22 2004
STATUS
approved