A308999 Irregular triangle T(n,k) read by rows: Lexicographically smallest marks on "perfect rulers" (as defined in A103294) of length n.

0, 0, 1, 0, 1, 2, 0, 1, 3, 0, 1, 2, 4, 0, 1, 2, 5, 0, 1, 4, 6, 0, 1, 2, 3, 7, 0, 1, 2, 5, 8, 0, 1, 2, 6, 9, 0, 1, 2, 3, 6, 10, 0, 1, 2, 3, 7, 11, 0, 1, 2, 3, 8, 12, 0, 1, 2, 6, 10, 13, 0, 1, 2, 3, 4, 9, 14, 0, 1, 2, 3, 4, 10, 15, 0, 1, 2, 3, 8, 12, 16

Offset: 0

Author

Bob Selcoe, Jul 04 2019

Keywords

• nonn
• tabf

Comments

Refer to A103294 for additional definitions, references and links.

All rulers (rows) start with mark 0 and end with mark n.

Row lengths are A103298(n) + 1.

Examples

			Triangle starts:
  0;
  0,  1;
  0,  1,  2;
  0,  1,  3;
  0,  1,  2,  4;
  0,  1,  2,  5;
  0,  1,  4,  6;
  0,  1,  2,  3,  7;
  0,  1,  2,  5,  8;
  0,  1,  2,  6,  9;
  0,  1,  2,  3,  6, 10;
  0,  1,  2,  3,  7, 11;
  0,  1,  2,  3,  8, 12;
  0,  1,  2,  6, 10, 13;
  0,  1,  2,  3,  4,  9, 14;
  0,  1,  2,  3,  4, 10, 15;
  0,  1,  2,  3,  8, 12, 16;
		

Links

• Peter Luschny, The perfect ruler pyramid
• Peter Luschny, Perfect rulers

Crossrefs

Cf. A103294, A103298.

Programs

Sage

def Partsum(T) :
    return [add([T[j] for j in range(i)]) for i in (0..len(T))]
def Ruler(L, S) :
    return map(Partsum, Compositions(L, length=S))
def isComplete(R) :
    S = Set([])
    L = len(R)-1
    for i in range(L,0,-1) :
        for j in (1..i) :
            S = S.union(Set([R[i]-R[i-j]]))
    return len(S) == R[L]
def CompleteRuler(L, S) :
    return list(filter(isComplete, Ruler(L, S)))
def PerfectRulers(L) :
    for i in (0..L) :
        R = CompleteRuler(L, i)
        if R: return R
    return []
def A308999list(L):
    for n in (0..L):
        print(PerfectRulers(n)[-1])
A308999list(16) # Peter Luschny, Aug 21 2019