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

Author

Gus Wiseman, Apr 02 2021

Keywords

• nonn

Examples

			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)
		

Links

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

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.

Programs

Mathematica

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

Formula

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

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