A276391 -id:A276391 - cp's OEIS frontend

A115716 A divide-and-conquer sequence.

1, 1, 3, 1, 3, 1, 11, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 683, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1, 3, 1, 171, 1, 3, 1, 11, 1, 3, 1, 43, 1, 3, 1, 11, 1

Offset: 0

Paul Barry, Jan 29 2006

			G.f. = 1 + x + 3*x^2 + x^3 + 3*x^4 + x^5 + 11*x^6 + x^7 + 3*x^8 + x^9 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..16381
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences , arXiv:math/0307027 [math.CO], 2003.

Partial sums are A032925.
Row sums of number triangle A115717.
Bisection: A276390.
Cf. A007583, A081294, A091090.
See A276391 for a closely related sequence.

a:= proc(n) option remember; `if`(n=0, 1,
      `if`(n::odd, 1, 4*a(n/2-1)-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 07 2016

a[n_] := a[n] = If[n == 0, 1, If[OddQ[n], 1, 4*a[n/2-1]-1]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 25 2017, after Alois P. Heinz *)

{a(n) = if( n<1, n==0, n%2, 1, 4 * a(n/2-1) - 1)}; /* Michael Somos, Sep 07 2016 */

The g.f. G(x) satisfies G(x)-4*x^2*G(x^2)=(1+2*x)/(1+x). - Argument and offset corrected by Bill Gosper, Sep 07 2016
G.f.: 1/(1-x) + Sum_{k>=0} ((4^k-0^k)/2) *x^(2^(k+1)-2) /(1-x^(2^k)). - corrected by R. J. Mathar, Sep 07 2016
a(n)=A007583(A091090(n+1)-1). - Adapted to new offset by R. J. Mathar, Sep 07 2016
a(0) = 1, a(2*n + 1) = 1 for n>=0. a(2*n + 2) = 4*a(n) - 1 for n>=0. - Michael Somos, Sep 07 2016

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

Original entry on oeis.org

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

Michael Day, May 22 2012

nonn

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

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.

Cf. A020985, A020991, A020986.

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

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);
}

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

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

cp's OEIS Frontend

A115716 A divide-and-conquer sequence.

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