A256494 -id:A256494 - cp's OEIS frontend

A258055 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.

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

Offset: 0

Armands Strazds, May 17 2015

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

			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

A. Strazds, The Golden Book [broken link]

Cf. A248646, A256494. See also A261300 for another version.

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 *)

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; }

A275536 Differences of the exponents of the adjacent distinct powers of 2 in the binary representation of n (with -1 subtracted from the least exponent present) are concatenated as decimal digits in reverse order.

Original entry on oeis.org

1, 2, 11, 3, 12, 21, 111, 4, 13, 22, 112, 31, 121, 211, 1111, 5, 14, 23, 113, 32, 122, 212, 1112, 41, 131, 221, 1121, 311, 1211, 2111, 11111, 6, 15, 24, 114, 33, 123, 213, 1113, 42, 132, 222, 1122, 312, 1212, 2112, 11112

Offset: 1

Armands Strazds, Aug 01 2016

A preferable representation is a sequence of arrays, since multi-digit items are possible: [1],[2],[1,1],[3],[1,2],[2,1],[1,1,1],[4],[1,3],[2,2],[1,1,2],[3,1],[1,2,1],[2,1,1],[1,1,1,1],[5],[1,4],[2,3],[1,1,3],[3,2],[1,2,2],[2,1,2],[1,1,1,2],[4,1],[1,3,1],[2,2,1],[1,1,2,1],[3,1,1],[1,2,1,1],[2,1,1,1],[1,1,1,1,1],[6],[1,5],[2,4],[1,1,4],[3,3],[1,2,3],[2,1,3],[1,1,1,3],[4,2],[1,3,2],[2,2,2],[1,1,2,2],[3,1,2],[1,2,1,2],[2,1,1,2],[1,1,1,1,2]. 0 is not allowed as a digit.
a(512) is the first term which cannot be expressed unambiguously in decimal. - Charles R Greathouse IV, Aug 02 2016
The first two terms which are equal (because of the ambiguity inherent in using decimal, or more generally any finite base) are a(3) = a(1024) = 11. a(3) corresponds to the array [1,1] while a(1024) corresponds to [11]. - Charles R Greathouse IV, Mar 19 2017

			5 = 2^2 + 2^0, so the representation is [2-0, 0-(-1)] = [2, 1] so a(5) = 12.
6 = 2^2 + 2^1, so the representation is [2-1, 1-(-1)] = [1, 2] so a(6) = 21.
18 = 2^4 + 2^1, so the representation is [4-1, 1-(-1)] = [3, 2] so a(18) = 23.

Charles R Greathouse IV, Table of n, a(n) for n = 1..511

Cf. A069800, A261300, A248646, A256494, A258055, A004086, A004719, A071160.

a(n)=my(v=List(),k); while(n, k=valuation(n,2)+1; n>>=k; listput(v,k)); fromdigits(Vec(v)) \\ Charles R Greathouse IV, Aug 02 2016

function dec2delta($k) {
  $p = -1;
  while ($k > 0) {
    $k -= $c = pow(2, floor(log($k, 2)));
    if ($p > -1) $d[] = $p - floor(log($c, 2));
    $p = floor(log($c, 2));
  }
  $d[] = $p + 1;
  return array_reverse($d);
}
function delta2dec($d) {
  $k = 0;
  $e = -1;
  foreach ($d AS $v) {
    if ($v > 0) {
      $e += $v;
      $k += pow(2, $e);
    }
  }
  return $k;
}

For n=1..511, a(n) = A004086(A004719(A071160(n))) [In other words, terms of A071160 with 0-digits deleted and the remaining digits reversed.] - Antti Karttunen, Sep 03 2016

cp's OEIS Frontend

A258055 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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