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A258055
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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.
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2
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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, 121, 11, 112, 111, 1111, 0, 5, 4, 41, 3, 32, 31, 311, 2, 23, 22, 221, 21, 212, 211, 2111, 1, 14, 13, 131, 12, 122, 121, 1211, 11, 113, 112, 1121, 111, 1112, 1111
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,6
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COMMENTS
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Originally called the "Golden Book's ZI-sequence" by the author.
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.
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
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LINKS
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EXAMPLE
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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.
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
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MATHEMATICA
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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 *)
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PROG
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(PHP)
function dec2zi ($d) {
$b = base_convert($d, 10, 2); $b = str_split($b);
$i = $z = 0; $r = "";
foreach($b as $v) {
if (!$v) {
$i++;
} else {
if ($i > 0) {
$r .= $i + $v; $i = 0;
} else {
if ($z > 0) {
$r .= $v; $z = 0;
}
$z++; }}}
return $r == "" ? 0 : $r; }
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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