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a(n)=f(a(n-1)+1,a(n-2)+1,a(n-3)), where f(x,y,z)=yz+zx+xy and (a(1),a(2),a(3))=(0,0,1).
2

%I #7 Nov 01 2022 13:01:30

%S 0,0,1,2,6,31,302,11706,3919531,47073195302,185055980038616706,

%T 8711901623180875318209094531,

%U 1612189902974799120179526719227635256224945302

%N a(n)=f(a(n-1)+1,a(n-2)+1,a(n-3)), where f(x,y,z)=yz+zx+xy and (a(1),a(2),a(3))=(0,0,1).

%C For a guide to related sequences, see A203761.

%t a[1] = 0; a[2] = 0; a[3] = 1;

%t a[n_] := SymmetricPolynomial[2, {1 + a[n - 1], 1 + a[n - 2], a[n - 3]}]

%t Table[a[n], {n, 1, 16}] (* A203901 *)

%t nxt[{a_,b_,c_}]:={b,c,Total[Times@@@Subsets[{a,b+1,c+1},{2}]]}; NestList[nxt,{0,0,1},15][[All,1]] (* _Harvey P. Dale_, Nov 01 2022 *)

%Y Cf. A203761.

%K nonn

%O 1,4

%A _Clark Kimberling_, Jan 07 2012