0,3

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.

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]

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.

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.

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}]

Cf. A000124, A072638.

Sequence in context: A136836 A136827 A136831 * A371047 A043681 A234472

Adjacent sequences: A204006 A204007 A204008 * A204010 A204011 A204012

Frank M Jackson, Jan 09 2012

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

approved