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A203162 (n-1)-st elementary symmetric function of the first n terms of (1,2,3,1,2,3,1,2,3,...).

%I A203162 #25 Sep 08 2022 08:46:01
%S A203162 1,3,11,17,40,132,168,372,1188,1404,3024,9504,10800,22896,71280,79056,
%T A203162 165888,513216,559872,1166400,3592512,3872448,8024832,24634368,
%U A203162 26313984,54307584,166281984,176359680,362797056,1108546560,1169012736
%N A203162 (n-1)-st elementary symmetric function of the first n terms of (1,2,3,1,2,3,1,2,3,...).
%H A203162 Clark Kimberling, <a href="/A203162/b203162.txt">Table of n, a(n) for n = 1..1000</a>
%H A203162 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,12,0,0,-36).
%F A203162 a(n) = 12*a(n-3)-36*a(n-6). - _Clark Kimberling_, Aug 18 2012
%F A203162 G.f.: x*(1 + 3*x + 11*x^2 + 5*x^3 + 4*x^4)/(1 - 6*x^3)^2. - _Clark Kimberling_, Aug 18 2012; corrected by _Georg Fischer_, May 10 2019
%e A203162 Let esf abbreviate "elementary symmetric function". Then
%e A203162 0th esf of {1}:  1;
%e A203162 1st esf of {1,2}:  1+2=3;
%e A203162 2nd esf of {1,2,3} is 1*2+1*3+2*3=11.
%t A203162 f[k_] := 1 + Mod[k + 2, 3]; t[n_] := Table[f[k], {k, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 40}] (* A203162 *)
%t A203162 Rest[CoefficientList[Series[x*(1 + 3*x + 11*x^2 + 5*x^3 + 4*x^4)/(1 - 6*x^3)^2, {x, 0, 30}], x]] (* _Vaclav Kotesovec_, May 10 2019 *)
%o A203162 (PARI) my(x='x+O('x^40)); Vec(x*(1+3*x+11*x^2+5*x^3+4*x^4)/(1-6*x^3)^2) \\ _G. C. Greubel_, May 10 2019
%o A203162 (Magma) I:=[1, 3, 11, 17, 40, 132]; [n le 6 select I[n] else 12*Self(n-3) -36*Self(n-6): n in [1..40]]; // _G. C. Greubel_, May 10 2019
%o A203162 (Sage) a=(x*(1+3*x+11*x^2+5*x^3+4*x^4)/(1-6*x^3)^2).series(x, 40).coefficients(x, sparse=False); a[1:] # _G. C. Greubel_, May 10 2019
%o A203162 (GAP) a:=[1, 3, 11, 17, 40, 132];; for n in [7..40] do a[n]:=12*a[n-1]-36*a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, May 10 2019
%Y A203162 Cf. A010882, A203160, A203161.
%K A203162 nonn,easy
%O A203162 1,2
%A A203162 _Clark Kimberling_, Dec 29 2011