A126606 Fixed point of transformation of the seed sequence {0,2}.

0, 2, 2, 4, 2, 6, 4, 6, 2, 8, 6, 10, 4, 10, 6, 8, 2, 10, 8, 14, 6, 16, 10, 14, 4, 14, 10, 16, 6, 14, 8, 10, 2, 12, 10, 18, 8, 22, 14, 20, 6, 22, 16, 26, 10, 24, 14, 18, 4, 18, 14, 24, 10, 26, 16, 22, 6, 20, 14, 22, 8, 18, 10, 12, 2, 14, 12, 22, 10, 28, 18, 26, 8, 30, 22, 36, 14, 34, 20, 26

Offset: 1

Author

Zak Seidov, Mar 13 2007

Keywords

• nonn

Comments

Start with s={0,2} If sum of two neighbor terms sum=s(i)+s(i+1) is even then insert the sum in between, otherwise insert abs(s(i)-s(i+1)); repeat the procedure.

{s(i),s(i+1)} => {s(i),s(i)+s(i+1), s(i+1)}, if s(i)+s(i+1) is even, otherwise {s(i),s(i+1)} => {s(i), abs(s(i)-s(i+1)), s(i+1)}.

Each row includes the previous one and then continues.

This sequence is analogous to Stern's diatomic series (A002487) but starting with 0,2 instead of 0,1. - Tom Edgar, May 08 2015

Examples

			Triangle begins:
  {0,2},
  {0,2,2},
  {0,2,2,4,2},
  {0,2,2,4,2,6,4,6,2},
  {0,2,2,4,2,6,4,6,2,8,6,10,4,10,6,8,2}.
		

Crossrefs

Cf. A002487.

Programs

Mathematica

s={0,2}; Do[t=s;ti=1; Do[If[EvenQ[su=s[[i]]+s[[i+1]]],t=Insert[t,su,i+ti], t=Insert[t,Abs[s[[i]]-s[[i+1]]],i+ti]];ti++,{i,Length[s]-1}];s=t,{8}];s
a[1]=0;a[2]=2;a[n_]:=If[EvenQ[n+1],a[(n+1)/2],a[(n)/2]+a[(n+2)/2]];Table[a[n],{n,100}] (* Vincenzo Librandi, May 09 2015 *)

Sage

def A126606(n):
    M = [2, 0]
    for b in n.bits():
        M[b] = M[0] + M[1]
    return M[1]
print([A126606(n) for n in (0..79)]) # Peter Luschny, Nov 28 2017

Formula

a(n) = 2 * A002487(n - 1). - Reikku Kulon, Oct 05 2008

a(1) = 0, a(2) = 2; for n>0: a(2n+1) = a(n+1) and a(2n) = a(n) + a(n+1). - Tom Edgar, May 08 2015