|
|
A373950
|
|
Number of integer compositions of n containing two adjacent ones and no other runs.
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
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))}
|
|
CROSSREFS
|
For any run (not just of ones) we have A003242.
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.
|
|
KEYWORD
|
nonn,new
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|