A204009 a(n) is a binary vector for selecting distinct terms from A000124 that when summed give n; it uses the greedy algorithm.

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 10000, 10001, 10010, 10011, 10100, 100000, 100001, 100010, 100011, 100100, 100101, 1000000, 1000001, 1000010, 1000011, 1000100, 1000101, 1000110, 10000000, 10000001, 10000010, 10000011, 10000100

Offset

0,3

Comments

a(n) is a binary vector for selecting terms from A000124 that when summed give n. It uses the greedy algorithm to select from multiple solutions.

Links

Table of n, a(n) for n=0..33.

Wikipedia, "Complete" sequence. [Wikipedia calls a sequence "complete" (sic) if every positive integer is a sum of distinct terms. This name is extremely misleading and should be avoided. - N. J. A. Sloane, May 20 2023]

Formula

a(n) x A000124 = n, where x is the inner product and the binary vector is in ascending powers of 2 with infinite trailing zeros.

Example

14 can be written as 7+4+2+1, i.e., 1111, or as 11+2+1, i.e., 10011, and the latter is chosen because it uses the greedy algorithm for selection.

Mathematica

complete[m_Integer] := (m(m+1)/2+1); gentable[n_Integer] := (m=n; ptable={0}; While[m!=0, (i=0; While[complete[i]<=m&&ptable[[i+1]]!=1, (AppendTo[ptable, 0]; i++)]; ptable[[i]]=1; m=m-complete[i-1])]; ptable); decimal[n_Integer] := (gentable[n]; Sum[2^(k-1)*ptable[[k]], {k, 1, Length[ptable]}]); Table[IntegerString[decimal[s], 2], {s, 0, 100}]

Crossrefs

Cf. A000124, A072638.

Sequence in context: A136836 A136827 A136831 * A371047 A043681 A234472

Adjacent sequences: A204006 A204007 A204008 * A204010 A204011 A204012

Keyword

nonn

Author

Frank M Jackson, Jan 09 2012

Extensions

Edited by N. J. A. Sloane, May 20 2023

Status

approved