|
|
A208354
|
|
Number of compositions of n with at most one even part.
|
|
11
|
|
|
1, 1, 2, 4, 7, 13, 23, 41, 72, 126, 219, 379, 653, 1121, 1918, 3272, 5567, 9449, 16003, 27049, 45636, 76866, 129267, 217079, 364057, 609793, 1020218, 1705036, 2846647, 4748101, 7912559, 13174889, 21919488, 36440646, 60538443, 100503667, 166744997, 276476129
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Conjecture: a(n) is the number of compositions of n if all the 1's are constrained to be in a single run; for example, a(7) counts the compositions 4,1,1,1 and 1,1,1,4 but not the compositions 1,4,1,1 and 1,1,4,1. - Gregory L. Simay, Sep 29 2018
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (x+1)*(x-1)^2/(x^2+x-1)^2.
a(n) = T(n+1) - T(n-1), where T(n) = ((2*n+3)*Fibonacci(n) - n*Fibonacci(n-1)) / 5 = A010049(n). - Gary Detlefs, Jan 19 2013
|
|
EXAMPLE
|
a(4) = 7: {4, 13, 31, 112, 121, 211, 1111}.
a(5) = 13: {5, 14, 41, 23, 32, 113, 131, 311, 1112, 1121, 1211, 2111, 11111}.
a(6) = 23: {6, 15, 51, 33, 114, 141, 411, 123, 132, 213, 231, 312, 321, 1113, 1131, 1311, 3111, 11112, 11121, 11211, 12111, 21111, 111111}.
|
|
MAPLE
|
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-2|1|2>>^n.
<<1, 1, 2, 4>>)[1, 1]:
seq(a(n), n=0..40);
|
|
MATHEMATICA
|
CoefficientList[Series[((-1 + x)^2 (1 + x))/(-1 + x + x^2)^2, {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)
|
|
PROG
|
(PARI) x='x+O('x^50); Vec((x+1)*(x-1)^2/(x^2+x-1)^2) \\ Altug Alkan, Oct 02 2018
(GAP) T:=n->((2*n+3)*Fibonacci(n)-n*Fibonacci(n-1))/5; a:=List([0..40], n->T(n+1)-T(n-1)); # Muniru A Asiru, Oct 28 2018
(Magma) I:=[1, 1, 2, 4]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Oct 29 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|