A319419 In binary expansion of n, delete one symbol from each run. Set a(n)=-1 if the result is the empty string.

-1, -1, -1, 1, 0, -1, 1, 3, 0, 0, -1, 1, 2, 1, 3, 7, 0, 0, 0, 1, 0, -1, 1, 3, 4, 2, 1, 3, 6, 3, 7, 15, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, -1, 1, 2, 1, 3, 7, 8, 4, 2, 5, 2, 1, 3, 7, 12, 6, 3, 7, 14, 7, 15, 31, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 2, 1, 3, 7, 0, 0

Offset: 0

Author

N. J. A. Sloane, Sep 21 2018

Keywords

• sign
• base
• hear

Comments

If the binary expansion of n is 1^b 0^c 1^d 0^e ..., then a(n) is the number whose binary expansion is 1^(b-1) 0^(c-1) 1^(d-1) 0^(e-1) .... Leading 0's are omitted, and if the result is the empty string, here we set a(n) = -1. See A318921 for a version which represents the empty string by 0.

Lenormand refers to this operation as planing ("raboter") the runs (or blocks) of the binary expansion.

A175046 expands the runs in a similar way, and a(A175046(n)) = A001477(n). - Andrew Weimholt, Sep 08 2018 (Comment copied from A318921.)

a(n) = -1 iff n in A000975.

Examples

			n / "planed" string / a(n)
0 e -1 (e = empty string)
1 e -1
10 e -1
11 1 1
100 0 0
101 e -1
110 1 1
111 11 3
1000 00 0
1001 0 0
1010 e -1
1011 1 1
1100 10 2
1101 1 1
1110 11 3
1111 111 7
10000 000 0
...
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..16384
• Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
• N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Crossrefs

Cf. A000975, A175046, A318921.

Programs

Maple

r:=proc(n) local t1, t2, L, len, i, j, k, b1;
if n <= 2 then return(-1); fi;
b1:=[]; t1:=convert(n, base, 2); L:=nops(t1); p:=1; len:=1;
for i from 2 to L do
t2:=t1[L+1-i];
if (t2=p) and (i1 then for j from 1 to len-1 do b1:=[op(b1), p]; od: fi;
   p:=t2; len:=1;
fi;               od;
if nops(b1)=0 then return(-1);
else k:=b1[1];
for i from 2 to nops(b1) do k:=2*k+b1[i]; od;
return(k);
fi;
end;
[seq(r(n), n=0..120)];

Python

from re import split
def A319419(n):
    s = ''.join(d[:-1] for d in split('(0+)|(1+)',bin(n)[2:]) if d not in {'','0','1',None})
    return -1 if s == '' else int(s,2) # Chai Wah Wu, Sep 24 2018

Python

from itertools import groupby
def a(n):
    s = "".join(k*(len(list(g))-1) for k, g in groupby(bin(n)[2:]))
    return int(s, 2) if s != "" else -1
print([a(n) for n in range(82)]) # Michael S. Branicky, Jun 01 2025