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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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27
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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
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OFFSET
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0,3
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COMMENTS
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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). - 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
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 1, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, ... - R. J. Mathar, Aug 10 2012
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LINKS
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FORMULA
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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
G.f.: 1/(1 - Sum_{k>=1} x^k*(1+x^k) ). - Joerg Arndt, Mar 09 2014
G.f.: 1/(1 - (x/(1-x)) - x^2/(1-x^2)) = 1/(1 - (x+2*x^2+x^3+2*x^4+x^5+2*x^6+...) ); in general 1/(1 - Sum_{j>=1} m(j)*x^j ) is the g.f. for compositions with m(k) sorts of part k. - Joerg Arndt, Feb 16 2015
a(n) = 3^((n-1)/2)*( 2*sqrt(3)*Fibonacci(n, 1/sqrt(3)) + Fibonacci(n, 1/sqrt(3)) ). - G. C. Greubel, Jan 15 2020
E.g.f.: 1/3 + (2/39)*exp(x/2)*(13*cosh((sqrt(13)*x)/2) + 2*sqrt(13)*sinh((sqrt(13)*x)/2)). - Stefano Spezia, Jan 15 2020
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EXAMPLE
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a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1).
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MAPLE
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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
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CoefficientList[Series[(1-x^2)/(1-x-3x^2), {x, 0, 35}], x] (* or *) Join[{1}, LinearRecurrence[{1, 3}, {1, 3}, 50]] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2011; typo fixed by Vincenzo Librandi, Jul 21 2013 *)
Table[Round[Sqrt[3]^(n-3)*(2*Sqrt[3]*Fibonacci[n+1, 1/Sqrt[3]] +Fibonacci[n, 1/Sqrt[3]])], {n, 0, 40}] (* G. C. Greubel, Jan 15 2020 *)
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PROG
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(Magma) I:=[1, 1, 3]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013
(Magma) R<x>:=PowerSeriesRing(Integers(), 33); Coefficients(R!( 1/(1-(x/(1-x))-x^2/(1-x^2)))); // Marius A. Burtea, Jan 15 2020
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x^2)/(1-x-3*x^2) ).list()
(GAP) a:=[1, 3];; for n in [3..40] do a[n]:=a[n-1]+3*a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Jan 15 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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