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1, then repeat 1,0.
11

%I #63 Apr 23 2024 11:13:31

%S 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,

%T 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,

%U 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

%N 1, then repeat 1,0.

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

%C 2-adic expansion of 1/3 (right to left): 1/3 = ...01010101010101011. - _Philippe Deléham_, Mar 24 2009

%C Also, with offset 0, parity of A036467(n-1). - _Omar E. Pol_, Mar 17 2015

%C 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

%H Andrei Asinowski, Cyril Banderier, and Valerie Roitner, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/several_patterns.pdf">Generating functions for lattice paths with several forbidden patterns</a>, (2019).

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).

%F G.f.: x*(1+x-x^2)/(1-x^2). - _Philippe Deléham_, Feb 08 2012

%F G.f.: x / (1 - x / (1 + x / (1 + x / (1 - x)))). - _Michael Somos_, Apr 02 2012

%F a(n) = A049711(n+2) mod 2. - _Ctibor O. Zizka_, Jan 28 2019

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

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

%t Prepend[Table[Mod[n + 1, 2], {n, 2, 60}], 1] (* _Michael De Vlieger_, Mar 17 2015 *)

%t PadRight[{1},120,{0,1}] (* _Harvey P. Dale_, Apr 23 2024 *)

%o (Haskell)

%o a135528 n = a135528_list !! (n-1)

%o a135528_list = concat $ iterate ([1,0] *) [1]

%o instance Num a => Num [a] where

%o fromInteger k = [fromInteger k]

%o (p:ps) + (q:qs) = p + q : ps + qs

%o ps + qs = ps ++ qs

%o (0:ps) * qs = 0 : ps * qs

%o (p:ps) * qs'@(q:qs) = p * q : ps * qs' + [p] * qs

%o _ * _ = []

%o -- _Reinhard Zumkeller_, Apr 02 2011

%Y Cf. A008578, A036467, A049711, A135416.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, based on a message from Guy Steele and _Don Knuth_, Mar 01 2008