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A052942
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Expansion of 1/((1+x)*(1-2*x+2*x^2-2*x^3)).
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10
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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
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OFFSET
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0,5
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COMMENTS
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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
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LINKS
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FORMULA
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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) = 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
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MAPLE
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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)):
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MATHEMATICA
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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 *)
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PROG
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(PARI) Vec( 1/(1-x-2*x^4) +O(x^66)) \\ Joerg Arndt, Aug 28 2013
(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
(Sage) (1/(1-x-2*x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 12 2019
(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
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CROSSREFS
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KEYWORD
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easy,nonn,changed
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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STATUS
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approved
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