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A208354 Number of compositions of n with at most one even part. 9
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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

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

LinearRecurrence[{2, 1, -2, -1}, {1, 1, 2, 4}, 40] (* Jean-Fran├žois Alcover, Feb 18 2017 *)

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

Cf. A010049, A211164.

Sequence in context: A239553 A319255 A136299 * A003116 A303666 A260917

Adjacent sequences:  A208351 A208352 A208353 * A208355 A208356 A208357

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 25 2012

STATUS

approved

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Last modified January 23 17:24 EST 2019. Contains 319399 sequences. (Running on oeis4.)