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

0, 1, 2, 5, 8, 9, 10, 21, 32, 33, 34, 37, 40, 41, 42, 85, 128, 129, 130, 133, 136, 137, 138, 149, 160, 161, 162, 165, 168, 169, 170, 341, 512, 513, 514, 517, 520, 521, 522, 533, 544, 545, 546, 549, 552, 553, 554, 597, 640, 641, 642, 645, 648, 649, 650, 661

Offset: 1

Author

Michael Day, May 22 2012

Keywords

• nonn

Comments

Brillhart and Morton derive an omega function for the largest values of k. This sequence appears to be given by a similar alpha function.

Links

• Michael Day, Table of n, a(n) for n = 1..10000
• J. Brillhart and P. Morton, A case study in mathematical research: the Golay-Rudin-Shapiro sequence, Amer. Math. Monthly, 103 (1996) 854-869.
• Kevin Ryde, Iterations of the Alternate Paperfolding Curve, see index GRScumulFirstN.

Crossrefs

Cf. A020985, A020991, A020986.

Programs

J

NB. J function on a vector
NB. Beware round-off errors on large arguments
NB. ok up to ~ 1e8
alphav =: 3 : 0
n   =. <: y
if.+/ ntlo=. n > 0 do.
n   =. ntlo#n
m   =. >.-: n
r   =. <.2^.m
f   =. <.3%~2+2^2*>:i.>./>:r
z   =. 0
mi  =. m
for_i. i.#f do.
  z   =. z + (i{f) * <.0.5 + mi =. mi%2
end.
nzer=. (+/ @: (0=>./\)@:|.)"1 @: #: m
ntlo #^:_1 z - (2|n) * <.-:nzer{f
else.
ntlo
end.
)
NB. eg    alphav 1 3 5 100 2 8 33

PARI

alpha(n)={
if(n<2, return(max(0,n-1)));
local(nm1=n-1,
      mi=m=ceil(nm1/2),
      r=floor(log(m)/log(2)),
i,fi,alpha=0,a);
forstep(i=1, 2*r+1, 2,
    mi/=2;
    fi=(1+2^i)\3;
alpha+=fi*floor(0.5+mi);
       );
alpha*=2;
if(nm1%2,   \\ adjust for even n
   a=factor(2*m)[1,2]-1;
alpha-= (1+2^(1+2*a))\3;
  );
return(alpha);
}

Formula

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.

Restating the formula above, a(n+1) - a(n) = A007583(A050605(n-1)) = A276391 with terms repeated. - John Keith, Mar 04 2021

Extensions

Minor edits by N. J. A. Sloane, Jun 06 2012