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A208354 Number of compositions of n with at most one even part. 10
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

a(n) = (2*(A099920(n-2)+A000045(n+2)) + A099920(n-1)+A000045(n+1)) / 5. - Yuchun Ji, Mar 21 2019

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 April 17 22:19 EDT 2021. Contains 343071 sequences. (Running on oeis4.)