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A052936
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Expansion of (1-x)(1-2x)/(1-5x+5x^2).
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1
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1, 2, 7, 25, 90, 325, 1175, 4250, 15375, 55625, 201250, 728125, 2634375, 9531250, 34484375, 124765625, 451406250, 1633203125, 5908984375, 21378906250, 77349609375, 279853515625, 1012519531250, 3663330078125, 13254052734375
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| For n>=0, a(n) is the number of generalized compositions of n+1 when there are 2^(i-1)+2 different types of i, (i=1,2,...). [From Milan R. Janjic (agnus(AT)blic.net), Sep 24 2010]
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 925
Index to sequences with linear recurrences with constant coefficients, signature (5,-5).
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FORMULA
| G.f.: (-1+x)*(-1+2*x)/(1-5*x+5*x^2)
Recurrence: {a(0)=1, a(1)=2, a(2)=7, 5*a(n)-5*a(n+1)+a(n+2)}
Sum(-1/5*(-1+_alpha)*_alpha^(-1-n), _alpha=RootOf(1-5*_Z+5*_Z^2))
The sequence beginning 2, 7, 25 ... has g.f. (2-3x)/(1-5x+5x^2), a(n)=(1-2/sqrt(5))(5/2-sqrt(5)/2)^n+(5/2+sqrt(5)/2)^n(1+2/sqrt(5)). It is the binomial transform of Fib(2n+3) and the second binomial transform of Fib(n+3). Also, its n-th term is the n-th term of the 3rd binomial transform of Fib(3n+3) divided by 2^n. - Paul Barry (pbarry(AT)wit.ie), Mar 23 2004
Binomial transform of convolution of Fib(2n+1) and (-1)^n. Binomial transform of Fib(n+1)^2. - Paul Barry (pbarry(AT)wit.ie), Sep 27 2004
a(n)=sum{k=0..n, C(n-1, k)(Fib(2n-2k)+Fib(2n-2k-1))} - Paul Barry (pbarry(AT)wit.ie), Jun 07 2005
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MAPLE
| spec := [S, {S=Sequence(Prod(Union(Sequence(Z), Sequence(Union(Z, Z))), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
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CROSSREFS
| Sequence in context: A070859 A048576 A018907 * A108152 A097613 A024482
Adjacent sequences: A052933 A052934 A052935 * A052937 A052938 A052939
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KEYWORD
| easy,nonn
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 06 2000
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