OFFSET
0,4
COMMENTS
Mirror of triangle A065120. See example. - Omar E. Pol, Oct 17 2013
a(n) is also the least part in the integer partition having viabin number n. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20. - Emeric Deutsch, Jul 24 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..65535 (first 10001 terms from Vincenzo Librandi)
FORMULA
a(2^k-1)=k; for any other n, a(n) = a(floor(n/2)).
a(n) = f(n, 0) with f(n, x) = if n < 2 then n + x else f([n/2], (x+1)*(n mod 2)). - Reinhard Zumkeller, Feb 02 2007
Conjecture: a(n) = w(n+1)*(w(n+1)-w(n)+1) - w(2^(w(n+1)+1)-n-1) for n>0, where w(n) = floor(log_2(n)), that is, A000523(n). - Velin Yanev, Dec 21 2016
a(n) = A360189(n-1,floor(log_2(n))). - Alois P. Heinz, Mar 06 2023
EXAMPLE
In binary : 14=1110 and there are 3 leading 1's, so a(14)=3.
From Omar E. Pol, Oct 17 2013: (Start)
Written as an irregular triangle with row lengths A011782 the sequence begins:
0;
1;
1,2;
1,1,2,3;
1,1,1,1,2,2,3,4;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6;
(End)
MAPLE
a := proc(n) if type(log[2](n+1), integer) then log[2](n+1) else a(floor((1/2)*n)) end if end proc: seq(a(n), n = 0 .. 200); # Emeric Deutsch, Jul 24 2017
# second Maple program:
b:= proc(n, t) `if`(n=0, t,
b(iquo(n, 2, 'm'), m*(t+1)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..127); # Alois P. Heinz, Mar 06 2023
MATHEMATICA
Join[{0}, Table[Length@First@Split@IntegerDigits[n, 2], {n, 30}]] (* Birkas Gyorgy, Mar 09 2011 *) (* adapted by Vincenzo Librandi, Dec 23 2016 *)
PROG
(PARI) a(n) = if(n==0, 0); b=binary(n+1); if(hammingweight(b) == 1, #b-1, a(n\2)) \\ David A. Corneth, Jul 24 2017
(PARI) a(n) = if(n==0, 0); my(b = binary(n), r = #b); for(i=2, #b, if(!b[i], return(i-1))); r \\ David A. Corneth, Jul 24 2017
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Benoit Cloitre, Feb 29 2004
EXTENSIONS
Edited and corrected by Franklin T. Adams-Watters, Apr 08 2006
Sequence had accidentally been shifted left by one step, which was corrected and term a(0)=0 added by Antti Karttunen, Jan 01 2007
STATUS
approved