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%I #29 Feb 21 2021 03:36:26
%S 0,0,0,1,0,2,1,11,0,3,2,21,1,12,11,111,0,4,3,31,2,22,21,211,1,13,12,
%T 121,11,112,111,1111,0,5,4,41,3,32,31,311,2,23,22,221,21,212,211,2111,
%U 1,14,13,131,12,122,121,1211,11,113,112,1121,111,1112,1111
%N Concatenation of the decimal representations of the lengths (increased by 1) of the runs of zeros between successive ones in the binary representation of n.
%C Originally called the "Golden Book's ZI-sequence" by the author.
%C The ZI-sequence is related to the binary numbers sequence with 10 ^ n substituted by the respective exponent increased by 1 (i.e., 10 as 2, 100 as 3, etc.) and the least significant bit discarded, e.g., binary 1011 converts to ZI 21.
%C a(n) = 0 when no successive ones exist in the binary representation of n, i.e., when n=0 and when n is a power of 2. - _Giovanni Resta_, Aug 31 2015
%H A. Strazds, <a href="http://www.zime.lv/book/?oeis">The Golden Book</a> [broken link]
%e Example for n=6: binary 110 => split into 10^m components: 1 (10^0) and 10 (10^1) => 1; the least significant bit, and thus the whole last component, here 10, is discarded.
%e 840 in binary is 1100101000. The runs of zeros between successive ones have length 0, 2 and 1, hence a(840) = 132. - _Giovanni Resta_, Aug 31 2015
%t a[0] = 0; a[n_] := FromDigits@ Flatten[ IntegerDigits /@ Most[ Length /@ (Split[ Flatten[ IntegerDigits[n, 2] /. 1 -> {1, 0}]][[2 ;; ;; 2]]) ]]; Table[a@ n, {n, 0, 100}] (* _Giovanni Resta_, Aug 31 2015 *)
%o (PHP)
%o function dec2zi ($d) {
%o $b = base_convert($d, 10, 2); $b = str_split($b);
%o $i = $z = 0; $r = "";
%o foreach($b as $v) {
%o if (!$v) {
%o $i++;
%o } else {
%o if ($i > 0) {
%o $r .= $i + $v; $i = 0;
%o } else {
%o if ($z > 0) {
%o $r .= $v; $z = 0;
%o }
%o $z++; }}}
%o return $r == "" ? 0 : $r; }
%Y Cf. A248646, A256494. See also A261300 for another version.
%K nonn,base,easy
%O 0,6
%A _Armands Strazds_, May 17 2015
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