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A105476
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Number of compositions of n when each even part can be of two kinds.
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22
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1, 1, 3, 6, 15, 33, 78, 177, 411, 942, 2175, 5001, 11526, 26529, 61107, 140694, 324015, 746097, 1718142, 3956433, 9110859, 20980158, 48312735, 111253209, 256191414, 589951041, 1358525283, 3128378406, 7203954255, 16589089473, 38200952238, 87968220657
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums of A105475.
Starting (1, 3, 6, 15,...) = sum of (n-1)-th row terms of triangle A140168. - Gary W. Adamson, May 10 2008
a(n) is also the number of compositions of n using 1's and 2's such that each run of like numbers can be grouped arbitrarily. For example, a(4) = 15 because 4 = (1)+(1)+(1)+(1) = (1+1)+(1)+(1) = (1)+(1+1)+(1) = (1)+(1)+(1+1) = (1+1)+(1+1) = (1+1+1)+(1) = (1)+(1+1+1) = (1+1+1+1) = (2)+(1)+(1) = (1)+(2)+(1) = (1)+(1)+(2) = (2)+(1+1) = (1+1)+(2) = (2)+(2) = (2+2). [From Martin J. Erickson (erickson(AT)truman.edu), Dec 09 2008]
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 69, 261, 321 and 324, lead to this sequence (without the first leading 1). For the corner squares these vectors lead to the companion sequence A006138. [Johannes W. Meijer, Aug 15 2010]
Inverse INVERT transform of the left shifted sequence gives A000034.
Eigensequence of the triangle
1,
2, 1,
1, 2, 1,
2, 1, 2, 1,
1, 2, 1, 2, 1,
2, 1, 2, 1, 2, 1,
1, 2, 1, 2, 1, 2, 1,
2, 1, 2, 1, 2, 1, 2, 1 ... [Paul Barry, Feb 10 2011]
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FORMULA
| G.f.: (1-x^2) / (1-x-3*x^2).
a(n) = a(n-1) + 3*a(n-2) for n>=3.
a(n) = ((2+sqrt(13))*(1+sqrt(13))^n-(2-sqrt(13))*(1-sqrt(13))^n)/(3*2^n*sqrt(13)) for n>0. - Bruno Berselli, May 24 2011
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EXAMPLE
| a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1).
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MAPLE
| G:=(1-z^2)/(1-z-3*z^2): Gser:=series(G, z=0, 35): 1, seq(coeff(Gser, z^n), n=1..33);
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MATHEMATICA
| CoefficientList[Series[(z^2 - 1)/(3*z^2 + z - 1), {z, 0, 100}], z] (* or *) Join[{1}, LinearRecurrence[{1, 3}, {1, 3}, 50]] (* From Vladimir Joseph Stephan Orlovsky, Jul 17 2011 *)
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CROSSREFS
| Cf. A105475, A006130, A105963.
Equals 3*A006138(n-2), n>1.
Sequence in context: A006961 A034740 A152167 * A000599 A063832 A006647
Adjacent sequences: A105473 A105474 A105475 * A105477 A105478 A105479
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
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