A052942 Expansion of 1/((1+x)*(1-2*x+2*x^2-2*x^3)).

1, 1, 1, 1, 3, 5, 7, 9, 15, 25, 39, 57, 87, 137, 215, 329, 503, 777, 1207, 1865, 2871, 4425, 6839, 10569, 16311, 25161, 38839, 59977, 92599, 142921, 220599, 340553, 525751, 811593, 1252791, 1933897, 2985399, 4608585, 7114167, 10981961

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 4, 3*a(n-4) equals the number of 3-colored compositions of n with all parts >= 4, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011

a(n+3) equals the number of ternary words of length n having at least 3 zeros between every two successive nonzero letters. - Milan Janjic, Mar 09 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 933
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,2).

Crossrefs

Column k=3 of A143453.

Programs

GAP

a:=[1,1,1,1];; for n in [5..40] do a[n]:=a[n-1]+2*a[n-4]; od; a; # G. C. Greubel, Jun 12 2019

Magma

I:=[1,1,1,1]; [n le 4 select I[n] else Self(n-1)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Mar 10 2015

Maple

spec := [S,{S=Sequence(Union(Z,Prod(Union(Z,Z),Z,Z,Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(add(binomial(n-3*k,k)*2^k, k=0..floor(n/3)), n=0..39); # Zerinvary Lajos, Apr 03 2007
with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S), S=Sequence(U, card >= 1), U=Sequence(Z, card >3)}, unlabeled]: seq(count(SeqSeqSeqL, size=j+4), j=0..39); # Zerinvary Lajos, Apr 04 2009
a := n -> `if`(n<9, [1, 1, 1, 1, 3, 5, 7, 9, 15][n+1], hypergeom([(1-n)/4,(2-n)/4,(3-n)/4,-n/4], [(1-n)/3,(2-n)/3,-n/3], -512/27)):
seq(simplify(a(n)),n=0..39); # Peter Luschny, Mar 09 2015

Mathematica

CoefficientList[Series[1/(1-x-2*x^4), {x,0,40}], x] (* Vincenzo Librandi, Mar 10 2015 *)
LinearRecurrence[{1,0,0,2},{1,1,1,1},50] (* Harvey P. Dale, Aug 17 2024 *)

PARI

Vec( 1/(1-x-2*x^4) + O(x^66)) \\ Joerg Arndt, Aug 28 2013

Sage

(1/(1-x-2*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019

Formula

G.f.: 1/(1-x-2*x^4).

a(n) = a(n-1) + 2*a(n-4), with a(1)=1, a(0)=1, a(2)=1, a(3)=1.

a(n) = Sum_{alpha=RootOf(-1+_Z+2*_Z^4)} (1/539)*(27 + 72*alpha^3 + 96*alpha^2 + 128*alpha)*alpha^(-1-n)).

a(n) = Sum_{k=0..floor(n/3)} A128099(n-2*k, k). - Johannes W. Meijer, Aug 28 2013

a(n) = hypergeom([(1-n)/4,(2-n)/4,(3-n)/4,-n/4],[(1-n)/3,(2-n)/3,-n/3],-512/27) for n>=9. - Peter Luschny, Mar 09 2015

Extensions

More terms from James Sellers, Jun 06 2000