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A052536
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Number of compositions of n when parts 1 and 2 are of two kinds.
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2
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1, 2, 6, 17, 49, 141, 406, 1169, 3366, 9692, 27907, 80355, 231373, 666212, 1918281, 5523470, 15904198, 45794313, 131859469, 379674209, 1093228314, 3147825473, 9063802210, 26098178316, 75146709475, 216376326215, 623030800329
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The g.f. for compositions of k_1 kinds of 1's, k_2 kinds of 2's, ..., k_j kinds of j's, ... is 1/(1-sum(j>=1, k_j * x^j )). [Joerg Arndt, Jul 06 2011]
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 467
Index to sequences with linear recurrences with constant coefficients, signature (3,0,-1).
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FORMULA
| G.f.: (1-x)/(1-3*x+x^3).
G.f.: 1/(1-(2*x+2*x^2+sum(j>=3, x^j )). [Joerg Arndt, Jul 06 2011]
a(n) = Sum( -1/9*(-2+r^2-r)*r^(-1-n), r=RootOf(1-3*x+x^3) )
a(0)=1, a(1)=2, a(2)=6, a(n)=3*a(n-1)-a(n-3) for n>=3. - Emeric Deutsch, Apr 10 2005
a(n) = left term in M^n * [1 0 0], where M = the 3X3 matrix [2 1 1 / 1 1 0 / 1 0 0]. Right term in M^n *[1 0 0] is a(n-1); middle term is A076264(n-1). - Gary W. Adamson, Sep 05 2005
3*a(n) = A123891(n+1). [From Jeffrey R. Goodwin, Jul 03 2011]
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EXAMPLE
| a(2)=6 because we have (2),(2'),(1,1),(1,1'),(1',1) and (1',1').
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MAPLE
| spec := [S, {S=Sequence(Union(Z, Prod(Z, Union(Z, Sequence(Z)))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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PROG
| (PARI) Vec((1-x)/(1-3*x+x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011
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CROSSREFS
| Row sums of A105478.
Cf. A105478, A076264.
Sequence in context: A077936 A077983 A036365 * A122100 A122099 A026165
Adjacent sequences: A052533 A052534 A052535 * A052537 A052538 A052539
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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
Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 10 2005
More terms from Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 05 2005
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