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A135247
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a(n) = 3*a(n-1) + 2*a(n-2) - 6*a(n-3).
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0
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1, 3, 11, 33, 103, 309, 935, 2805, 8431, 25293, 75911, 227733, 683263, 2049789, 6149495, 18448485, 55345711, 166037133, 498111911, 1494335733, 4483008223, 13449024669, 40347076055, 121041228165, 363123688591, 1089371065773, 3268113205511, 9804339616533
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of partitions of n into parts 1 (in three colors) and 2 (in two colors) where the order of colors matters. For example, the a(2)=11 such partitions (using parts 1, 1', 1'', 2, and 2') are 2, 2', 1+1, 1+1', 1+1'', 1'+1, 1'+1', 1'+1'', 1''+1, 1''+1', 1''+1''. For such partitions where the order of colors does not matter see A002624. - Joerg Arndt, Jan 18 2024
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LINKS
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FORMULA
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a(n) = 3*a(n-1) + A077957(n) for n >= 1.
(End)
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MAPLE
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seq(coeff(series(1/(1-3*x-2*x^2+6*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 20 2019
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MATHEMATICA
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LinearRecurrence[{3, 2, -6}, {1, 3, 11}, 30] (* Harvey P. Dale, Jun 27 2015 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ G. C. Greubel, Nov 20 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // G. C. Greubel, Nov 20 2019
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
(GAP) a:=[1, 3, 11];; for n in [4..30] do a[n]:=3*a[n-1]+2*a[n-2]-6*a[n-3]; od; a; # G. C. Greubel, Nov 20 2019
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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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EXTENSIONS
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Dropped two leading terms = 0. - Joerg Arndt, Jan 18 2024
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STATUS
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approved
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