A135528 1, then repeat 1,0.

1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Author

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Keywords

• nonn
• easy

Comments

This is Guy Steele's sequence GS(2, 1) (see A135416).

2-adic expansion of 1/3 (right to left): 1/3 = ...01010101010101011. - Philippe Deléham, Mar 24 2009

Also, with offset 0, parity of A036467(n-1). - Omar E. Pol, Mar 17 2015

Appears to be the Gilbreath transform of 1,2,3,5,7,11,13,... (A008578). (This is essentially the same as the Gilbreath conjecture, see A036262.) - N. J. A. Sloane, May 08 2023

Examples

			G.f. = x + x^2 + x^4 + x^6 + x^8 + x^10 + x^12 + x^14 + x^16 + x^18 + x^20 + ...
		

Links

• Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A008578, A036467, A049711, A135416.

Programs

Haskell

a135528 n = a135528_list !! (n-1)
a135528_list = concat $ iterate ([1,0] *) [1]
instance Num a => Num [a] where
fromInteger k = [fromInteger k]
   (p:ps) + (q:qs) = p + q : ps + qs
   ps + qs         = ps ++ qs
   (0:ps) * qs         = 0 : ps * qs
   (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs
    *                = []
-- Reinhard Zumkeller, Apr 02 2011

Maple

GS(2,1,200); [see A135416].

Mathematica

Prepend[Table[Mod[n + 1, 2], {n, 2, 60}], 1] (* Michael De Vlieger, Mar 17 2015 *)
PadRight[{1},120,{0,1}] (* Harvey P. Dale, Apr 23 2024 *)

Formula

G.f.: x*(1+x-x^2)/(1-x^2). - Philippe Deléham, Feb 08 2012

G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x)))). - Michael Somos, Apr 02 2012

a(n) = A049711(n+2) mod 2. - Ctibor O. Zizka, Jan 28 2019