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A212591 a(n) is the smallest value of k for which A020986(k) = n.

%I A212591 #40 Dec 12 2021 22:11:09
%S A212591 0,1,2,5,8,9,10,21,32,33,34,37,40,41,42,85,128,129,130,133,136,137,
%T A212591 138,149,160,161,162,165,168,169,170,341,512,513,514,517,520,521,522,
%U A212591 533,544,545,546,549,552,553,554,597,640,641,642,645,648,649,650,661
%N A212591 a(n) is the smallest value of k for which A020986(k) = n.
%C A212591 Brillhart and Morton derive an omega function for the largest values of k. This sequence appears to be given by a similar alpha function.
%H A212591 Michael Day, <a href="/A212591/b212591.txt">Table of n, a(n) for n = 1..10000</a>
%H A212591 J. Brillhart and P. Morton, <a href="http://www.maa.org/programs/maa-awards/writing-awards/a-case-study-in-mathematical-research-the-golay-rudin-shapiro-sequence">A case study in mathematical research: the Golay-Rudin-Shapiro sequence</a>, Amer. Math. Monthly, 103 (1996) 854-869.
%H A212591 Kevin Ryde, <a href="http://user42.tuxfamily.org/alternate/index.html">Iterations of the Alternate Paperfolding Curve</a>, see index GRScumulFirstN.
%F A212591 a(2*n-1) - a(2*n-2) = (2^(2*k+1)+1)/3 and a(2*n) - a(2*n-1) = (2^(2*k+1)+1)/3 with a(0) = a(1) = 0, where n = (2^k)*(2*m-1) for some integers k >= 0 and m > 0.
%F A212591 Restating the formula above, a(n+1) - a(n) = A007583(A050605(n-1)) = A276391 with terms repeated. - _John Keith_, Mar 04 2021
%o A212591 (PARI)
%o A212591 alpha(n)={
%o A212591 if(n<2, return(max(0,n-1)));
%o A212591 local(nm1=n-1,
%o A212591       mi=m=ceil(nm1/2),
%o A212591       r=floor(log(m)/log(2)),
%o A212591 i,fi,alpha=0,a);
%o A212591 forstep(i=1, 2*r+1, 2,
%o A212591     mi/=2;
%o A212591     fi=(1+2^i)\3;
%o A212591 alpha+=fi*floor(0.5+mi);
%o A212591        );
%o A212591 alpha*=2;
%o A212591 if(nm1%2,   \\ adjust for even n
%o A212591    a=factor(2*m)[1,2]-1;
%o A212591 alpha-= (1+2^(1+2*a))\3;
%o A212591   );
%o A212591 return(alpha);
%o A212591 }
%o A212591 (J)
%o A212591 NB. J function on a vector
%o A212591 NB. Beware round-off errors on large arguments
%o A212591 NB. ok up to ~ 1e8
%o A212591 alphav =: 3 : 0
%o A212591 n   =. <: y
%o A212591 if.+/ ntlo=. n > 0 do.
%o A212591 n   =. ntlo#n
%o A212591 m   =. >.-: n
%o A212591 r   =. <.2^.m
%o A212591 f   =. <.3%~2+2^2*>:i.>./>:r
%o A212591 z   =. 0
%o A212591 mi  =. m
%o A212591 for_i. i.#f do.
%o A212591   z   =. z + (i{f) * <.0.5 + mi =. mi%2
%o A212591 end.
%o A212591 nzer=. (+/ @: (0=>./\)@:|.)"1 @: #: m
%o A212591 ntlo #^:_1 z - (2|n) * <.-:nzer{f
%o A212591 else.
%o A212591 ntlo
%o A212591 end.
%o A212591 )
%o A212591 NB. eg    alphav 1 3 5 100 2 8 33
%Y A212591 Cf. A020985, A020991, A020986.
%K A212591 nonn
%O A212591 1,3
%A A212591 _Michael Day_, May 22 2012
%E A212591 Minor edits by _N. J. A. Sloane_, Jun 06 2012