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

1, 3, 11, 17, 40, 132, 168, 372, 1188, 1404, 3024, 9504, 10800, 22896, 71280, 79056, 165888, 513216, 559872, 1166400, 3592512, 3872448, 8024832, 24634368, 26313984, 54307584, 166281984, 176359680, 362797056, 1108546560, 1169012736

Offset

1,2

Links

Clark Kimberling, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,12,0,0,-36).

Formula

a(n) = 12*a(n-3)-36*a(n-6). - Clark Kimberling, Aug 18 2012

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

Example

Let esf abbreviate "elementary symmetric function". Then
0th esf of {1}:  1;
1st esf of {1,2}:  1+2=3;
2nd esf of {1,2,3} is 1*2+1*3+2*3=11.

Mathematica

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 *)
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 *)

Prog

(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
(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
(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
(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

Crossrefs

Cf. A010882, A203160, A203161.

Sequence in context: A302872 A023865 A024592 * A240084 A100567 A270225

Adjacent sequences: A203159 A203160 A203161 * A203163 A203164 A203165

Keyword

nonn,easy

Author

Clark Kimberling, Dec 29 2011

Status

approved