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%I #40 Jan 20 2024 09:19:05
%S 1,3,11,33,103,309,935,2805,8431,25293,75911,227733,683263,2049789,
%T 6149495,18448485,55345711,166037133,498111911,1494335733,4483008223,
%U 13449024669,40347076055,121041228165,363123688591,1089371065773,3268113205511,9804339616533
%N a(n) = 3*a(n-1) + 2*a(n-2) - 6*a(n-3).
%C This sequence interleaves A016133 and 3*A016133, see formulas. - _Mathew Englander_, Jan 08 2024
%C 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
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,-6).
%F G.f.: 1/((1-3*x)*(1-2*x^2)). - _G. C. Greubel_, Oct 04 2016
%F From _Mathew Englander_, Jan 08 2024: (Start)
%F a(n) = A010684(n) * A016133(floor(n/2)).
%F a(n) = 3*a(n-1) + A077957(n) for n >= 1.
%F a(n) = (A000244(n+2) - A164073(n+3))/7.
%F (End)
%p 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
%t LinearRecurrence[{3,2,-6},{1,3,11},30] (* _Harvey P. Dale_, Jun 27 2015 *)
%o (PARI) my(x='x+O('x^30)); Vec(1/(1-3*x-2*x^2+6*x^3)) \\ _G. C. Greubel_, Nov 20 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/(1-3*x-2*x^2+6*x^3) )); // _G. C. Greubel_, Nov 20 2019
%o (Sage)
%o def A135247_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/(1-3*x-2*x^2+6*x^3) ).list()
%o A135247_list(30) # _G. C. Greubel_, Nov 20 2019
%o (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
%Y Cf. A016133.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Feb 15 2008
%E More terms from _Harvey P. Dale_, Jun 27 2015
%E Dropped two leading terms = 0. - _Joerg Arndt_, Jan 18 2024
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