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

%I A308999 #17 Feb 27 2020 11:46:33
%S A308999 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,
%T A308999 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,
%U A308999 14,0,1,2,3,4,10,15,0,1,2,3,8,12,16
%N A308999 Irregular triangle T(n,k) read by rows: Lexicographically smallest marks on "perfect rulers" (as defined in A103294) of length n.
%C A308999 Refer to A103294 for additional definitions, references and links.
%C A308999 All rulers (rows) start with mark 0 and end with mark n.
%C A308999 Row lengths are A103298(n) + 1.
%H A308999 Peter Luschny, <a href="http://www.luschny.de/math/rulers/rulerpyramid.html">The perfect ruler pyramid</a>
%H A308999 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/PerfectRulers">Perfect rulers</a>
%e A308999 Triangle starts:
%e A308999   0;
%e A308999   0,  1;
%e A308999   0,  1,  2;
%e A308999   0,  1,  3;
%e A308999   0,  1,  2,  4;
%e A308999   0,  1,  2,  5;
%e A308999   0,  1,  4,  6;
%e A308999   0,  1,  2,  3,  7;
%e A308999   0,  1,  2,  5,  8;
%e A308999   0,  1,  2,  6,  9;
%e A308999   0,  1,  2,  3,  6, 10;
%e A308999   0,  1,  2,  3,  7, 11;
%e A308999   0,  1,  2,  3,  8, 12;
%e A308999   0,  1,  2,  6, 10, 13;
%e A308999   0,  1,  2,  3,  4,  9, 14;
%e A308999   0,  1,  2,  3,  4, 10, 15;
%e A308999   0,  1,  2,  3,  8, 12, 16;
%o A308999 (Sage)
%o A308999 def Partsum(T) :
%o A308999     return [add([T[j] for j in range(i)]) for i in (0..len(T))]
%o A308999 def Ruler(L, S) :
%o A308999     return map(Partsum, Compositions(L, length=S))
%o A308999 def isComplete(R) :
%o A308999     S = Set([])
%o A308999     L = len(R)-1
%o A308999     for i in range(L,0,-1) :
%o A308999         for j in (1..i) :
%o A308999             S = S.union(Set([R[i]-R[i-j]]))
%o A308999     return len(S) == R[L]
%o A308999 def CompleteRuler(L, S) :
%o A308999     return list(filter(isComplete, Ruler(L, S)))
%o A308999 def PerfectRulers(L) :
%o A308999     for i in (0..L) :
%o A308999         R = CompleteRuler(L, i)
%o A308999         if R: return R
%o A308999     return []
%o A308999 def A308999list(L):
%o A308999     for n in (0..L):
%o A308999         print(PerfectRulers(n)[-1])
%o A308999 A308999list(16) # _Peter Luschny_, Aug 21 2019
%Y A308999 Cf. A103294, A103298.
%K A308999 nonn,tabf
%O A308999 0,6
%A A308999 _Bob Selcoe_, Jul 04 2019