

A281185


a(0)=0, a(1)=1, a(2)=0; thereafter, a(2n)=a(n)+a(n+1) for n >= 2, a(2n+1)=a(n) for n >= 1.


2



0, 1, 0, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 2, 2, 1, 2, 1, 3, 1, 2, 2, 3, 0, 5, 3, 4, 2, 3, 2, 3, 1, 3, 2, 4, 1, 4, 3, 3, 1, 4, 2, 5, 2, 3, 3, 5, 0, 8, 5, 7, 3, 6, 4, 5, 2, 5, 3, 5, 2, 4, 3, 4, 1, 5, 3, 6, 2, 5, 4, 5, 1, 7, 4, 6, 3, 4, 3, 5, 1, 6, 4, 7, 2, 7, 5, 5, 2, 6, 3, 8, 3, 5, 5, 8, 0, 13, 8, 12, 5
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OFFSET

0,7


COMMENTS

A "bow" sequence. The bow sequences are a family of recursive sequences defined to have the flipped recursion from the Stern sequence A002487 (called bow for the opposite end of the boat from the stern). The bow sequences require two initial conditions: a(1)=alpha, a(2)=beta. We also define a(0)=0, although it does not enter into the recursion.
The bow sequences then follow the recursion a(2n)=a(n)+a(n+1) for n at least 2, and a(2n+1)=a(n). This particular bow sequence has initial conditions a(1)=0, a(2)=1 and (along with the sequence A106345 with initial conditions a(1)=1, a(2)=0) is of particular importance when studying the general bow sequences.


LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..25000
M. Dennison, A Sequence Related to the Stern Sequence, Ph.D. dissertation, University of Illinois at UrbanaChampaign, 2010.
Melissa Dennison, On Properties of the General Bow Sequence, J. Int. Seq., Vol. 22 (2019), Article 19.2.7.


EXAMPLE

a(3)=a(1)=1, a(4)=a(2)+a(3)=0+1=1, a(5)=a(2)=0.


MAPLE

f:=proc(n) option remember;
if n=0 then 0
elif n=1 then 1
elif n=2 then 0
else
if n mod 2 = 0 then f(n/2)+f(1+n/2) else f((n1)/2) fi;
fi;
end;
[seq(f(n), n=0..150)]; # N. J. A. Sloane, Apr 26 2017


MATHEMATICA

b[0]=0; b[1]=1; b[2]=0; b[n_?EvenQ]:=b[n]=b[n/2]+b[n/2+1]; b[n_?OddQ]:=b[n]=b[(n1)/2]


CROSSREFS

Cf. A002487, A106345.
Sequence in context: A035180 A163819 A301734 * A260683 A092673 A243842
Adjacent sequences: A281182 A281183 A281184 * A281186 A281187 A281188


KEYWORD

nonn,look,easy


AUTHOR

Melissa Dennison, Apr 12 2017


EXTENSIONS

Edited by N. J. A. Sloane, Apr 26 2017


STATUS

approved



