A342194 Number of strict compositions of n with equal differences, or strict arithmetic progressions summing to n.

1, 1, 1, 3, 3, 5, 7, 7, 7, 13, 11, 11, 17, 13, 15, 25, 17, 17, 29, 19, 23, 35, 25, 23, 39, 29, 29, 45, 33, 29, 55, 31, 35, 55, 39, 43, 65, 37, 43, 65, 51, 41, 77, 43, 51, 85, 53, 47, 85, 53, 65, 87, 61, 53, 99, 67, 67, 97, 67, 59, 119, 61, 71, 113, 75, 79, 123, 67, 79, 117

Offset

0,4

Links

Table of n, a(n) for n=0..69.

Wikipedia, Arithmetic progression.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Formula

a(n > 0) = A175342(n) - A000005(n) + 1.

a(n > 0) = 2*A049988(n) - 2*A000005(n) + 1 = 2*A049982(n) + 1.

Example

The a(1) = 1 through a(9) = 13 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)    (8)    (9)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)  (1,7)  (1,8)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)  (2,6)  (2,7)
                          (3,2)  (4,2)    (3,4)  (3,5)  (3,6)
                          (4,1)  (5,1)    (4,3)  (5,3)  (4,5)
                                 (1,2,3)  (5,2)  (6,2)  (5,4)
                                 (3,2,1)  (6,1)  (7,1)  (6,3)
                                                        (7,2)
                                                        (8,1)
                                                        (1,3,5)
                                                        (2,3,4)
                                                        (4,3,2)
                                                        (5,3,1)

Mathematica

Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&], SameQ@@Differences[#]&]], {n, 0, 30}]

Crossrefs

Strict compositions in general are counted by A032020.

The unordered version is A049980.

The non-strict version is A175342.

A000203 adds up divisors.

A000726 counts partitions with alternating parts unequal.

A003242 counts anti-run compositions.

A224958 counts compositions with alternating parts unequal.

A342343 counts compositions with alternating parts strictly decreasing.

A342495 counts compositions with constant quotients.

A342527 counts compositions with alternating parts equal.

Cf. A000009, A001522, A002843, A049988, A062968, A070211, A114921, A325545, A325557, A342496, A342515, A342522.

Sequence in context: A110246 A070543 A377419 * A196372 A168053 A222657

Adjacent sequences: A342191 A342192 A342193 * A342195 A342196 A342197

Keyword

nonn

Author

Gus Wiseman, Apr 02 2021

Status

approved