A373950 Number of integer compositions of n containing two adjacent ones and no other runs.

0, 0, 1, 0, 2, 4, 5, 14, 26, 46, 92, 176, 323, 610, 1145, 2108, 3912, 7240, 13289, 24418, 44778, 81814, 149356, 272222, 495144, 899554, 1632176, 2957332, 5352495, 9677266, 17477761, 31536288, 56852495, 102403134, 184302331, 331452440, 595659234, 1069742760

Offset

0,5

Comments

Also the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) results in a composition of n-1.

Links

John Tyler Rascoe, Table of n, a(n) for n = 0..1000

Formula

a(n>0) = A373949(n,n-1).

G.f.: x/((1-x)^2 * (1 - Sum_{i>0} (x^i/(1+x^i)))^2). - John Tyler Rascoe, Jul 02 2024

Example

The a(0) = 0 through a(7) = 14 compositions:
  .  .  (11)  .  (112)  (113)   (114)   (115)
                 (211)  (311)   (411)   (511)
                        (1121)  (1131)  (1123)
                        (1211)  (1311)  (1132)
                                (2112)  (1141)
                                        (1411)
                                        (2113)
                                        (2311)
                                        (3112)
                                        (3211)
                                        (11212)
                                        (12112)
                                        (21121)
                                        (21211)

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-1&]], {n, 0, 10}]

Prog

(PARI)
A_x(N)={my(x='x+O('x^N), h=x/((1+x)^2*(1-sum(i=1, N, (x^i /(1+x^i))))^2)); concat([0, 0], Vec(h))}
A_x(40) \\ John Tyler Rascoe, Jul 02 2024

Crossrefs

For any run (not just of ones) we have A003242.

Subdiagonal of A373949.

These compositions are ranked by A373956.

A003242 counts compressed compositions.

A114901 counts compositions with no isolated parts.

A116861 counts partitions by compressed sum, by compressed length A116608.

A333755 counts compositions by compressed length (number of runs).

A373948 represents the run-compression transformation.

Cf. A106356, A238130, A238279, A238343, A240085, A285981, A333213, A333489, A373951, A373952.

Sequence in context: A121410 A002959 A306919 * A295031 A178436 A370959

Adjacent sequences: A373947 A373948 A373949 * A373951 A373952 A373953

Keyword

nonn

Author

Gus Wiseman, Jun 28 2024

Extensions

a(26) onwards from John Tyler Rascoe, Jul 02 2024

Status

approved