A342495 Number of compositions of n with constant (equal) first quotients.

1, 1, 2, 4, 5, 6, 8, 10, 10, 11, 12, 12, 16, 16, 18, 20, 19, 18, 22, 22, 24, 28, 24, 24, 30, 27, 30, 30, 34, 30, 38, 36, 36, 36, 36, 40, 43, 40, 42, 46, 48, 42, 52, 46, 48, 52, 48, 48, 56, 55, 54, 54, 58, 54, 60, 58, 64, 64, 60, 60, 72, 64, 68, 74, 69, 72, 72

Offset: 0

Author

Gus Wiseman, Mar 17 2021

Keywords

• nonn

Comments

The first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).

Examples

			The composition (1,2,4,8) has first quotients (2,2,2) so is counted under a(15).
The composition (4,5,6) has first quotients (5/4,6/5) so is not counted under a(15).
The a(1) = 1 through a(7) = 10 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (12)   (13)    (14)     (15)      (16)
             (21)   (22)    (23)     (24)      (25)
             (111)  (31)    (32)     (33)      (34)
                    (1111)  (41)     (42)      (43)
                            (11111)  (51)      (52)
                                     (222)     (61)
                                     (111111)  (124)
                                               (421)
                                               (1111111)
		

Links

• Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.
• Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Crossrefs

The version for differences instead of quotients is A175342.

The unordered version is A342496, ranked by A342522.

The strict unordered version is A342515.

The distinct version is A342529.

A000005 counts constant compositions.

A000009 counts strictly increasing (or strictly decreasing) compositions.

A000041 counts weakly increasing (or weakly decreasing) compositions.

A003238 counts chains of divisors summing to n - 1 (strict: A122651).

A167865 counts strict chains of divisors > 1 summing to n.

Cf. A002843, A003242, A008965, A048004, A059966, A074206, A167606, A253249, A318991, A318992, A325557, A342528.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Divide@@@Partition[#,2,1]&]],{n,0,15}]

Formula

a(n > 0) = 2*A342496(n) - A000005(n).