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 A052536 Number of compositions of n when parts 1 and 2 are of two kinds. 8
 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; text; internal format)
 OFFSET 0,2 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 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..2177 Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 467 Index entries for linear recurrences with constant coefficients, signature (3,0,-1). 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 3 X 3 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). - Jeffrey R. Goodwin, Jul 03 2011 EXAMPLE a(2)=6 because we have (2),(2'),(1,1),(1,1'),(1',1) and (1',1'). MAPLE spec := [S, {S=Sequence(Union(Z, Prod(Z, Union(Z, Sequence(Z)))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA a[0] = 1; a[1] = 2; a[2] = 6; a[n_] := a[n] = 3*a[n-1] - a[n-3]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Jun 12 2013, after Emeric Deutsch *) PROG (PARI) Vec((1-x)/(1-3*x+x^3)+O(x^99)) \\ Charles R Greathouse IV, Nov 20 2011 CROSSREFS Row sums of A105478. Cf. A105478, A076264. Sequence in context: A036365 A299162 A244400 * A122100 A122099 A026165 Adjacent sequences: A052533 A052534 A052535 * A052537 A052538 A052539 KEYWORD easy,nonn AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 EXTENSIONS More terms from James A. Sellers, Jun 06 2000 Edited by Emeric Deutsch, Apr 10 2005 More terms from Gary W. Adamson, Sep 05 2005 STATUS approved

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Last modified November 30 01:31 EST 2022. Contains 358431 sequences. (Running on oeis4.)