A325677 Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.

1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 2, 1, 6, 6, 1, 6, 8, 1, 8, 18, 1, 8, 16, 1, 10, 30, 4, 1, 10, 34, 14, 1, 12, 48, 28, 1, 12, 48, 42, 1, 14, 72, 76, 1, 14, 72, 100, 1, 16, 96, 160, 8, 1, 16, 98, 190, 8, 1, 18, 126, 284, 40, 1, 18, 128, 316, 70

Offset

1,4

Comments

Also the number of length-k compositions of n such that every restriction to a subinterval has a different sum. A composition of n is a finite sequence of positive integers summing to n.

Links

Table of n, a(n) for n=1..67.

Eric Weisstein's World of Mathematics, Golomb Ruler.

Example

Triangle begins:
   1
   1
   1   2
   1   2
   1   4
   1   4   2
   1   6   6
   1   6   8
   1   8  18
   1   8  16
   1  10  30   4
   1  10  34  14
   1  12  48  28
   1  12  48  42
   1  14  72  76
   1  14  72 100
   1  16  96 160   8
   1  16  98 190   8
   1  18 126 284  40
   1  18 128 316  70
Row n = 8 counts the following rulers:
  {0,8}  {0,1,8}  {0,1,3,8}
         {0,2,8}  {0,1,5,8}
         {0,3,8}  {0,1,6,8}
         {0,5,8}  {0,2,3,8}
         {0,6,8}  {0,2,7,8}
         {0,7,8}  {0,3,7,8}
                  {0,5,6,8}
                  {0,5,7,8}
and the following compositions:
  (8)  (17)  (125)
       (26)  (143)
       (35)  (152)
       (53)  (215)
       (62)  (251)
       (71)  (341)
             (512)
             (521)

Mathematica

DeleteCases[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], UnsameQ@@ReplaceList[#, {___, s__, ___}:>Plus[s]]&]], {n, 15}, {k, n}], 0, {2}]

Crossrefs

Row sums are A169942.

Row lengths are A325678(n) = A143824(n + 1) - 1.

Column k = 2 is A052928.

Column k = 3 is A325686.

Rightmost column is A325683.

Cf. A000079, A007318, A103295, A108917, A143823, A325676, A325679, A325687.

Sequence in context: A238552 A257523 A067044 * A343411 A287477 A231473

Adjacent sequences: A325674 A325675 A325676 * A325678 A325679 A325680

Keyword

nonn,tabf

Author

Gus Wiseman, May 13 2019

Status

approved