A246588 Run Length Transform of S(n) = wt(n) = 0,1,1,2,1,2,2,3,1,... (cf. A000120).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1

Offset

0,8

Comments

The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Maple

A000120 := 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: wt := A000120;
ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n, base, 2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
   if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
   elif out1 = 0 and t1[i] = 1 then c:=c+1;
   elif out1 = 1 and t1[i] = 0 then c:=c;
   elif out1 = 0 and t1[i] = 0 then lis:=[c, op(lis)]; out1:=1; c:=0;
   fi;
   if i = L1 and c>0 then lis:=[c, op(lis)]; fi;
                   od:
a:=mul(wt(i), i in lis);
ans:=[op(ans), a];
od:
ans;

Mathematica

f[n_] := DigitCount[n, 2, 1]; Table[Times @@ (f[Length[#]]&)  /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 100}] (* Jean-François Alcover, Jul 11 2017 *)

Prog

(Haskell)
import Data.List (group)
a246588 = product . map (a000120 . length) .
          filter ((== 1) . head) . group . a030308_row
-- Reinhard Zumkeller, Feb 13 2015, Sep 05 2014
(Python)
from operator import mul
from functools import reduce
from re import split
def A246588(n):
    return reduce(mul, (bin(len(d)).count('1') for d in split('0+', bin(n)[2:]) if d)) if n > 0 else 1 # Chai Wah Wu, Sep 07 2014
(Sage) # uses[RLT from A246660]
A246588_list = lambda len: RLT(lambda n: sum(Integer(n).digits(2)), len)
A246588_list(88) # Peter Luschny, Sep 07 2014

Crossrefs

Cf. A000120.

Cf. A167489, A030308.

Run Length Transforms of other sequences: A071053, A227349, A246595, A246596, A246660, A246661, A246674.

Sequence in context: A376632 A373082 A119849 * A026492 A182284 A139551

Adjacent sequences: A246585 A246586 A246587 * A246589 A246590 A246591

Keyword

nonn

Author

N. J. A. Sloane, Sep 05 2014

Status

approved