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%I #9 Sep 08 2022 08:45:28
%S 1,1,2,3,6,10,17,30,52,90,156,271,470,815,1414,2453,4255,7381,12804,
%T 22211,38529,66836,115940,201120,348881,605201,1049837,1821143,
%U 3159121,5480100,9506282,16490465,28605867,49622350,86079461,149321296
%N Number of sequences with terms 1, 2 or 3 summing to n with no three consecutive 1's.
%H G. C. Greubel, <a href="/A123908/b123908.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,2,2,1).
%F a(n) = a(n-2) + 2*a(n-3) + 2*a(n-4) + a(n-5).
%F G.f.: (1 + x + x^2)/(1 - x^2 - 2*x^3 - 2*x^4 - x^5). - _Chai Wah Wu_, May 28 2016
%e a(4) = 6 as 3 + 1, 1 + 3, 2 + 2, 1 + 1 + 2, 1 + 2 + 1 and 2 + 1 + 1 (but not 1 + 1 + 1 + 1).
%p a[0]:=1; a[1]:=1; a[2]:=2; a[3]:=3; a[4]:=6; for n from 5 to 45 do a[n] := a[n-2] +2*a[n-3] +2*a[n-4] +a[n-5] end do; seq(a[n], n = 0 .. 40); # modified by _G. C. Greubel_, Aug 06 2019
%p seq(coeff(series((1+x+x^2)/(1-x^2-2*x^3-2*x^4-x^5), x, n+1), x, n), n = 0 .. 40); # _G. C. Greubel_, Aug 06 2019
%t LinearRecurrence[{0,1,2,2,1}, {1,1,2,3,6}, 40] (* _G. C. Greubel_, Aug 06 2019 *)
%o (PARI) my(x='x+O('x^40)); Vec((1+x+x^2)/(1-x^2-2*x^3-2*x^4-x^5)) \\ _G. C. Greubel_, Aug 06 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x+x^2)/(1-x^2-2*x^3-2*x^4-x^5) )); // _G. C. Greubel_, Aug 06 2019
%o (Sage) ((1+x+x^2)/(1-x^2-2*x^3-2*x^4-x^5)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 06 2019
%o (GAP) a:=[1,1,2,3,6];; for n in [6..40] do a[n]:=a[n-2]+2*a[n-3]+ 2*a[n-4]+a[n-5]; od; a; # _G. C. Greubel_, Aug 06 2019
%K easy,nonn
%O 0,3
%A _Joel B. Lewis_, Oct 28 2006