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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,...). 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 30 05:36 EST 2022. Contains 358431 sequences. (Running on oeis4.)