OFFSET
1,3
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.
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
PROG
(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);
}
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Day, May 22 2012
EXTENSIONS
Minor edits by N. J. A. Sloane, Jun 06 2012
STATUS
approved