A245196 Write n>=1 as either n=2^k-2^r with 0 <= r <= k-1, in which case a(2^k-2^r)=wt(k-r-1), or as n=2^k-2^r+j with 2 <= r <= k-1, 1 <= j < 2^r-1, in which case a(2^k-2^r+j)=a(j)*wt(k-r-1) (where wt(i) = A000120(i)).

0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0

Offset: 1

Author

N. J. A. Sloane, Jul 25 2014

Keywords

• nonn
• tabf

Comments

Other sequences defined by a recurrence of this class (see the Formula and Maple sections) include A245180, A245195, A048896, A245536, A038374.

Examples

			May be arranged into blocks of lengths 1,2,4,8,...:
0,
0, 1,
0, 0, 1, 1,
0, 0, 0, 0, 1, 0, 1, 2,
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 2,
...
		

Crossrefs

Cf. A000120, A160239, A245180, A245195, A014081, A048896, A038374, A245536, A245537, A245538, A245539, A048896, A245547.

Programs

Maple

Maple code for this sequence:
wt := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
G:=[seq(wt(n),n=0..30)];
m:=1;
f:=proc(n) option remember; global m,G; local k,r,j,np;
   k:=1+floor(log[2](n)); np:=2^k-n;
   if np=1 then r:=0; j:=0; else r:=1+floor(log[2](np-1)); j:=2^r-np; fi;
   if j=0 then G[k-r]; else m*G[k-r]*f(j); fi;
end;
[seq(f(n),n=1..120)];
# Maple code for the general recurrence:
G:=[seq(wt(n),n=0..30)]; # replace this by a list G=[G(0), G(1), G(2), ...], remembering that you have to tell Maple G[1] to get G(0), G[2] to get G(1), etc.
m:=1; # replace this by the correct multiplier
f:=proc(n) option remember; global m,G; local k,r,j,np;
   k:=1+floor(log[2](n)); np:=2^k-n;
   if np=1 then r:=0; j:=0; else r:=1+floor(log[2](np-1)); j:=2^r-np; fi;
   if j=0 then G[k-r-1+1]; else m*G[k-r-1+1]*f(j); fi;
end;
[seq(f(n),n=1..120)];
# If G(n) = wt(n) and m=1 we get the present sequence
# If G(n) = A083424(n) and m=1 we get A245537
# If G(n) = A083424(n) and m=2 we get A245538
# If G(n) = A083424(n) and m=4 we get A245539
# If G(n) = A083424(n) and m=8 we get A245180 (and presumably A160239)
# If G(n) = n (n>=0) and m=1 we get A245536
# If G(n) = n+1 (n>=0) and m=1 we get A038374
# If G(n) = (n+1)(n+2)/2 (n>=0) and m=1 we get A245541
# If G(n) = (n+1)(n+2)/2 (n>=0) and m=2 we get A245547
# If G(n) = 2^n (n>=0) and m=1 we get A245195 (= 2^A014081)
# If G(n) = 2^n (n>=0) and m=2 we get A048896

Formula

This is an example of a class of sequences defined by the following recurrence.

We first choose a sequence G = [G(0), G(1), G(2), G(3), ...], which are the terms that will appear at the ends of the blocks: a(2^k-1) = G(k-1), and we also choose a parameter m (the "multiplier"). Then the recurrence (this defines a(1), a(2), a(3), ...) is:

a(2^k-2^r)=G(k-r-1) if 0 <= r <= k-1, a(2^k-2^r+j)=m*a(j)*G(k-r-1) if 2 <= r <= k-1, 1 <= j < 2^r-1.

To help apply the recurrence, here are the values of k,r,j for the first few values of n (if n=2^k-2^r we set j=0, although it is not used):

n: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

k: 1 2 2 3 3 3 3 4 4 4 4 4 4 4 4

r: 0 1 0 2 2 1 0 3 3 3 3 2 2 1 0

j: 0 0 0 0 1 0 0 0 1 2 3 0 1 0 0

--------------------------------------------------

n: 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

k: 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

r: 4 4 4 4 4 4 4 4 3 3 3 3 2 2 1 0

j: 0 1 2 3 4 5 6 7 0 1 2 3 0 1 0 0

--------------------------------------------------

In the present example G(n) = wt(n) and m=1.