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

0,3

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).

Links

Table of n, a(n) for n=0..66.

Eric Weisstein's World of Mathematics, Logarithmically Concave Sequence.

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Formula

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

Example

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)

Mathematica

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

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.

Sequence in context: A015844 A114318 A169956 * A035500 A080653 A115836

Adjacent sequences: A342492 A342493 A342494 * A342496 A342497 A342498

Keyword

nonn

Author

Gus Wiseman, Mar 17 2021

Status

approved