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 A103294 Triangle T, read by rows: T(n,k) = number of complete rulers with length n and k segments (n >= 0, k >= 0). 21
 1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, 0, 4, 4, 1, 0, 0, 0, 2, 9, 5, 1, 0, 0, 0, 0, 12, 14, 6, 1, 0, 0, 0, 0, 8, 27, 20, 7, 1, 0, 0, 0, 0, 4, 40, 48, 27, 8, 1, 0, 0, 0, 0, 0, 38, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 30, 134, 166, 110, 44, 10, 1, 0, 0, 0, 0, 0, 14, 166, 311, 277, 154, 54, 11, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS If n=k then T(n,k)=1. A sparse ruler, or simply a ruler, is a strict increasing finite sequence of nonnegative integers starting from 0 called marks. A segment of a ruler is the space between two adjacent marks. The number of segments is the number of marks - 1. A ruler is complete if the set of all distances it can measure is {1,2,3,...,k} for some integer k>=1. A ruler is perfect if it is complete and no complete ruler with the same length possesses less marks. A ruler is optimal if it is perfect and no perfect ruler with the same number of segments has a greater length. The 'empty ruler' with length n=0 is considered perfect and optimal. REFERENCES G. S. Bloom and S. W. Golomb, Numbered complete graphs, unusual rulers, and assorted applications. Theory and Applications of Graphs, Lecture Notes in Math. 642, (1978), 53-65. R. K. Guy, Modular difference sets and error correcting codes. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter C10, pp. 181-183, 2004. J. C. P. Miller, Difference bases: Three problems in additive number theory, pp. 299-322 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971. LINKS G. S. Bloom and S. W. Golomb, Applications of numbered undirected graphs, Proc. IEEE 65 (1977), 562-570. Peter Luschny, Perfect and Optimal Rulers Peter Luschny, Table of Counts Peter Luschny, Perfect rulers Eric Weisstein's World of Mathematics, Perfect Rulers B. Wichmann, A note on restricted difference bases, J. Lond. Math. Soc. 38 (1963), 465-466. EXAMPLE Rows begin: [1], [0,1], [0,0,1], [0,0,2,1], [0,0,0,3,1], [0,0,0,4,4,1], [0,0,0,2,9,5,1], [0,0,0,0,12,14,6,1], [0,0,0,0,8,27,20,7,1], ... a(19)=T(5,4)=4 counts the complete rulers with length 5 and 4 segments: {[0,2,3,4,5],[0,1,3,4,5],[0,1,2,4,5],[0,1,2,3,5]} MATHEMATICA marks[n_, k_] := Module[{i}, i[0] = 0; iter = Sequence @@ Table[{i[j], i[j - 1] + 1, n - k + j - 1}, {j, 1, k}]; Table[Join[{0}, Array[i, k], {n}],      iter // Evaluate] // Flatten[#, k - 1]&]; completeQ[ruler_List] := Range[ruler[[-1]]] == Sort[ Union[ Flatten[ Table[ ruler[[i]] - ruler[[j]], {i, 1, Length[ruler]}, {j, 1, i - 1}]]]]; rulers[n_, k_] := Select[marks[n, k - 1], completeQ]; T[n_, n_] = 1; T[_, 0] = 0; T[n_, k_] := Length[rulers[n, k]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Quiet (* Jean-François Alcover, Jul 05 2019 *) PROG (Sage) 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 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 CompleteRuler(L, S) :     return tuple(filter(isComplete, Ruler(L, S))) for n in (0..8):     print([len(CompleteRuler(n, k)) for k in (0..n)]) # Peter Luschny, Jul 05 2019 CROSSREFS Row sums give A103295. Column sums give A103296. The first nonzero entries in the rows give A103300. The last nonzero entries in the columns give A103299. The row numbers of the last nonzero entries in the columns give A004137. Cf. A103295 through A103301, A004137, A212661. Sequence in context: A064287 A196389 A128206 * A198379 A079680 A037855 Adjacent sequences:  A103291 A103292 A103293 * A103295 A103296 A103297 KEYWORD nonn,tabl AUTHOR Peter Luschny, Feb 28 2005 EXTENSIONS Typo in data corrected by Jean-François Alcover, Jul 05 2019 STATUS approved

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Last modified December 1 03:31 EST 2020. Contains 338833 sequences. (Running on oeis4.)