A091072 Numbers whose odd part is of the form 4k+1. The bit to the left of the least significant bit of each term is unset.

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 21, 25, 26, 29, 32, 33, 34, 36, 37, 40, 41, 42, 45, 49, 50, 52, 53, 57, 58, 61, 64, 65, 66, 68, 69, 72, 73, 74, 77, 80, 81, 82, 84, 85, 89, 90, 93, 97, 98, 100, 101, 104, 105, 106, 109, 113, 114, 116, 117, 121, 122, 125, 128, 129

Offset

1,2

Comments

Either of form 2a(m) or 4k+1, k >= 0, 0 < m < n.

A000265(a(n)) is an element of A016813.

a(n) such that A038189(a(n)) = 0.

Numbers n such that kronecker(n, m) = kronecker(m, n) for all m. - Michael Somos, Sep 24 2005

The Dragon curve A014577 (but changing the offset to 1: (1, 1, 0, 1, 1, 0, 0, 1, 1, 1, ...) = the characteristic function of A091072. - Gary W. Adamson, Apr 11 2010

A014707(a(n) + 1) = 0. - Reinhard Zumkeller, Sep 28 2011

A055975(a(n)) > 0. - Reinhard Zumkeller, Apr 28 2012

Also indices of 1 in A034947. - Jianing Song, Apr 24 2021

The terms in the sequence are the same as the terms in the odd columns of the table in A135764 with headings 4k+1: (1, 5, 9, 13...). A014577(n) = 1 if n is in that set, but A014577(n) = 0 if n is in the set of even columns in the A135764 table. - Gary W. Adamson, May 29 2021

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 TurnLeft, with a(n) = TurnLeft(n-1).

J. E. S. Socolar and J. M. Taylor, An aperiodic hexagonal tile, arXiv:1003.4279 [math.CO], 2010.

Example

x + 2*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 9*x^6 + 10*x^7 + 13*x^8 + 16*x^9 + ...

Mathematica

Select[ Range[129], EvenQ[ (#/2^IntegerExponent[#, 2] - 1)/2 ] & ] (* Jean-François Alcover, Feb 16 2012, after Pari *)

Prog

(PARI) for(n=1, 200, if(((n/2^valuation(n, 2)-1)/2)%2==0, print1(n", ")))
(PARI) {a(n) = local(m, c); if( n<1, 0, c=1; m=1; while( c<n, m++; if( ((m / 2^valuation( m, 2) - 1) / 2)%2==0, c++)); m)} /* Michael Somos, Sep 24 2005 */
(PARI) a(n) = if(n=2*n-2, my(t=1); forstep(i=logint(n, 2), 0, -1, if(bittest(n, i)==t, n--; t=!t))); n+1; \\ Kevin Ryde, Mar 21 2021
(Haskell)
import Data.List (elemIndices)
a091072 n = a091072_list !! (n-1)
a091072_list = map (+ 1) $ elemIndices 0 a014707_list
-- Reinhard Zumkeller, Sep 28 2011

Crossrefs

Complement of A091067.

Cf. A014577 (characteristic function), A034947, A106841 (first of triplet), A088742 (first differences), A339597.

Sequence in context: A084581 A121996 A269176 * A001481 A248151 A245226

Adjacent sequences: A091069 A091070 A091071 * A091073 A091074 A091075

Keyword

nonn,easy

Author

Ralf Stephan, Feb 22 2004

Status

approved