cp's OEIS Frontend

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

%I A319419 #28 Jun 01 2025 13:14:36
%S A319419 -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,
%T A319419 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,
%U A319419 31,0,0,0,1,0,0,1,3,0,0,0,1,2,1,3,7,0,0
%N A319419 In binary expansion of n, delete one symbol from each run. Set a(n)=-1 if the result is the empty string.
%C A319419 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.
%C A319419 Lenormand refers to this operation as planing ("raboter") the runs (or blocks) of the binary expansion.
%C A319419 A175046 expands the runs in a similar way, and a(A175046(n)) = A001477(n). - _Andrew Weimholt_, Sep 08 2018 (Comment copied from A318921.)
%C A319419 a(n) = -1 iff n in A000975.
%H A319419 N. J. A. Sloane, <a href="/A319419/b319419.txt">Table of n, a(n) for n = 0..16384</a>
%H A319419 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
%H A319419 N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides.</a> (Mentions this sequence)
%e A319419 n / "planed" string / a(n)
%e A319419 0 e -1 (e = empty string)
%e A319419 1 e -1
%e A319419 10 e -1
%e A319419 11 1 1
%e A319419 100 0 0
%e A319419 101 e -1
%e A319419 110 1 1
%e A319419 111 11 3
%e A319419 1000 00 0
%e A319419 1001 0 0
%e A319419 1010 e -1
%e A319419 1011 1 1
%e A319419 1100 10 2
%e A319419 1101 1 1
%e A319419 1110 11 3
%e A319419 1111 111 7
%e A319419 10000 000 0
%e A319419 ...
%p A319419 r:=proc(n) local t1, t2, L, len, i, j, k, b1;
%p A319419 if n <= 2 then return(-1); fi;
%p A319419 b1:=[]; t1:=convert(n, base, 2); L:=nops(t1); p:=1; len:=1;
%p A319419 for i from 2 to L do
%p A319419 t2:=t1[L+1-i];
%p A319419 if (t2=p) and (i<L) then len:=len+1;
%p A319419 else # run ended
%p A319419    if (i = L) and (t2=p) then len:=len+1; fi;
%p A319419    if len>1 then for j from 1 to len-1 do b1:=[op(b1), p]; od: fi;
%p A319419    p:=t2; len:=1;
%p A319419 fi;               od;
%p A319419 if nops(b1)=0 then return(-1);
%p A319419 else k:=b1[1];
%p A319419 for i from 2 to nops(b1) do k:=2*k+b1[i]; od;
%p A319419 return(k);
%p A319419 fi;
%p A319419 end;
%p A319419 [seq(r(n), n=0..120)];
%o A319419 (Python)
%o A319419 from re import split
%o A319419 def A319419(n):
%o A319419     s = ''.join(d[:-1] for d in split('(0+)|(1+)',bin(n)[2:]) if d not in {'','0','1',None})
%o A319419     return -1 if s == '' else int(s,2) # _Chai Wah Wu_, Sep 24 2018
%o A319419 (Python)
%o A319419 from itertools import groupby
%o A319419 def a(n):
%o A319419     s = "".join(k*(len(list(g))-1) for k, g in groupby(bin(n)[2:]))
%o A319419     return int(s, 2) if s != "" else -1
%o A319419 print([a(n) for n in range(82)]) # _Michael S. Branicky_, Jun 01 2025
%Y A319419 Cf. A000975, A175046, A318921.
%K A319419 sign,base,hear
%O A319419 0,8
%A A319419 _N. J. A. Sloane_, Sep 21 2018