Displaying 1-10 of 18 results found.
Erroneous version of A004137 given in the reference.
+20
1
3, 6, 9, 13, 17, 23, 29, 36, 43, 50, 59, 60, 79, 90, 101, 112, 123, 138
REFERENCES
J.-C. Bermond, Graceful graphs, radio antennae and French windmills, pp. 18-37 of R. J. Wilson, editor, Graph Theory and Combinatorics. Pitman, London, 1978.
Number of perfect rulers with length n.
+10
30
1, 1, 1, 2, 3, 4, 2, 12, 8, 4, 38, 30, 14, 6, 130, 80, 32, 12, 500, 326, 150, 66, 18, 4, 944, 460, 166, 56, 12, 6, 2036, 890, 304, 120, 20, 10, 2, 2678, 974, 362, 100, 36, 4, 2, 4892, 2114, 684, 238, 68, 22, 4, 16318, 6350, 2286, 836, 330, 108, 24, 12, 31980, 12252
COMMENTS
For definitions, references and links related to complete rulers see A103294. The values for n = 208-213 are 22,0,0,0,4,4 according to Arch D. Robison. The values for 199-207 are not yet known. - Peter Luschny, Feb 20 2014, Jun 28 2019 Zero values at 135, 136, 149, 150, 151, 164, 165, 166, 179, 180, 181, 195, 196, 209, 210, 211. - Ed Pegg Jr, Jun 23 2019 [These values were found by Arch D. Robison, see links. Peter Luschny, Jun 28 2019] Zero values at 135, 136, 149, 150, 151, 164, 165, 166, 179, 180, 181, 195, 196 have been replaced with correct values using an additional mark.
A lower bound for 209 is 62, for 210 is 16, and for 211 is 204.
The verified value for 212 and for 213 is 4. (End)
EXAMPLE
a(5)=4 counts the perfect rulers with length 5, {[0,1,3,5],[0,2,4,5],[0,1,2,5],[0,3,4,5]}.
CROSSREFS
Cf. A004137 (Maximal number of edges in a graceful graph on n nodes).
Triangle T, read by rows: T(n,k) = number of complete rulers with length n and k segments (n >= 0, k >= 0).
+10
22
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
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.
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]];
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
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.
Size of smallest subset S of N={0,1,2,...,n} such that S-S=N, where S-S={abs(i-j) | i,j in S}.
+10
10
1, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16
COMMENTS
It is easy to show that a(n+1) must be no larger than a(n)+1. Problem: Can a(n+1) ever be smaller than a(n)?
Problem above solved in A103300. a(137) smaller than a(136). Except for initial term, round(sqrt(3*n + 9/4)) up to n=51. See A308766 for divergences up to n=213. See A326499 for a list of best known solutions. From Ed Pegg Jr, Jun 23 2019: (Start)
Minimal marks for a sparse ruler of length n.
Minimal vertices in a graceful graph with n edges. (End)
EXAMPLE
a(10)=6 since all integers in {0,1,2...10} are differences of elements of {0,1,2,3,6,10}, but not of any 5-element set.
a(17)=7 since all integers in {0,1,2...17} are differences of elements of {0,1,8,11,13,15,17}, but not of any 6-element set.
In other words, {0,1,8,11,13,15,17} is a restricted difference basis w.r.t. A004137(7)=17.
MATHEMATICA
Prepend[Table[Round[Sqrt[3*n+9/4]]+If[MemberQ[ A308766, n], 1, 0], {n, 1, 213}], 1]
Maximum length of a perfect Wichmann ruler with n segments.
+10
7
3, 6, 9, 12, 15, 22, 29, 36, 43, 50, 57, 68, 79, 90, 101, 112, 123, 138, 153, 168, 183, 198, 213, 232, 251, 270, 289, 308, 327, 350, 373, 396, 419, 442, 465, 492, 519, 546, 573, 600, 627, 658, 689, 720, 751, 782, 813, 848, 883, 918, 953, 988, 1023, 1062, 1101, 1140, 1179, 1218, 1257, 1300, 1343, 1386, 1429
FORMULA
a(n) = ( n^2 - (mod(n,6)-3)^2 ) / 3 + n.
G.f.: x^2*(3 + 4*x^5 - 3*x^6) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = 2*a(n-1) - a(n-2) + a(n-6) - 2*a(n-7) + a(n-8) for n>9.
(End)
PROG
(PARI) a(n) = n + (n^2 - (n%6 - 3)^2)/3; \\ Michel Marcus, Jul 14 2017 (Python)
Numbers k such that the minimal mark in a length k sparse ruler is round(sqrt(9 + 12*k)/2) + 1.
+10
6
51, 59, 69, 113, 124, 125, 135, 136, 139, 149, 150, 151, 164, 165, 166, 179, 180, 181, 195, 196, 199, 209, 210, 211
COMMENTS
Other sparse rulers in the range length 1 to 213 have round(sqrt(9 + 12*k)/2) minimal marks.
Minimal vertices in k-edge graceful graph = minimal marks in length k sparse ruler.
Minimal marks can be derived from A004137 and using zero-count values in A103300. Conjecture: Minimal marks k - round(sqrt(9 + 12*k)/2) is always 0 or 1.
Maximal number of edges in a b^{hat} graceful graph with n nodes.
(Formerly M2528)
+10
5
0, 1, 3, 6, 9, 13, 18, 24, 29
COMMENTS
A graph with e edges is 'b^{hat} graceful' if its nodes can be labeled with distinct nonnegative integers so that, if each edge is labeled with the absolute difference between the labels of its endpoints, then the e edges have the distinct labels 1, 2, ..., e.
Equivalently, maximum m for which there's a difference basis with respect to m with n elements. A 'difference basis w.r.t. m' is a set of integers such that every integer from 1 to m is a difference between two elements of the set.
Miller's paper gives these lower bounds for the 11 terms from a(9) to a(19): 29,37,45,51,61,70,79,93,101,113,127. (Bermond's paper gives these as exact values, but quotes Miller as their source.)
REFERENCES
J.-C. Bermond, Graceful graphs, radio antennae and French windmills, pp. 18-37 of R. J. Wilson, editor, Graph Theory and Combinatorics. Pitman, London, 1978.
R. K. Guy, Unsolved Problems in Number Theory, Sect. C10.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
EXAMPLE
a(7)=18: Label the 7 nodes 0,6,9,10,17,22,24 and include all edges except those from 0 to 22, from 0 to 24 and from 17 to 24. {0,6,9,10,17,22,24} is a difference basis w.r.t. 18.
Number of complete rulers with n segments.
+10
4
1, 1, 3, 10, 38, 175, 885, 5101, 32080, 219569, 1616882, 12747354, 106948772, 950494868
COMMENTS
For definitions, references and links related to complete rulers see A103294.
FORMULA
a(n) = Sum_{i=n.. A004137(n+1)} T(i, n) where T is the A103294 triangle.
EXAMPLE
a(2)=3 counts the complete rulers with 2 segments, {[0,1,2],[0,1,3],[0,2,3]}.
PROG
Link to FORTRAN program given in A103295.
CROSSREFS
Cf. A103301 (perfect rulers with n segments), A103299 (optimal rulers with n segments).
Number of optimal rulers with n segments (n>=0).
+10
4
1, 1, 2, 2, 4, 6, 12, 4, 6, 2, 2, 4, 12, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4
COMMENTS
For definitions, references and links related to complete rulers see A103294.
EXAMPLE
a(5)=6 counts the optimal rulers with 5 segments, {[0,1,6,9,11,13], [0,2,4,7,12,13], [0,1,4,5,11,13], [0,2,8,9,12,13], [0,1,2,6,10,13], [0,3,7,11,12,13]}.
CROSSREFS
Cf. A103296 (Complete rulers with n segments), A103301 (Perfect rulers with n segments).
EXTENSIONS
Terms a(20)-a(24) proved by exhaustive search by Fabian Schwartau, Yannic Schröder, Lars Wolf, Joerg Schoebel, Feb 22 2021
Number of perfect rulers with n segments (n>=0).
+10
4
1, 1, 3, 9, 24, 88, 254, 1064, 1644, 3382, 4156, 8022, 26264, 52012, 25434, 8506, 5632, 6224, 12330, 34224, 108854, 103156, 75992, 86560, 69084
COMMENTS
For definitions, references and links related to complete rulers see A103294.
EXAMPLE
a(3)=9 counts the perfect rulers with 3 segments, {[0,1,2,4],[0,2,3,4], [0,1,3,4],[0,1,3,5],[0,2,4,5],[0,1,2,5],[0,3,4,5],[0,1,4,6],[0,2,5,6]}.
EXTENSIONS
Terms a(19)-a(24) found by exhaustive search by Fabian Schwartau, Yannic Schröder, Lars Wolf, Joerg Schoebel, Feb 23 2021
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