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A379104 a(n) = third elementary symmetric function of the first n distinct Fibonacci numbers.

%I A379104 #8 Dec 23 2024 10:11:22
%S A379104 6,61,389,2066,9962,45594,202344,881859,3801171,16275292,69399116,
%T A379104 295177196,1253532482,5318285553,22550198601,95580699774,405034367814,
%U A379104 1716140731030,7270703692340,30801852323495,130485697292231,552764498063256,2341595675572344
%N A379104 a(n) = third elementary symmetric function of the first n distinct Fibonacci numbers.
%C A379104 Conjecture: for k >=1, the k-th elementary symmetric function (esf) of first n distinct Fibonacci numbers (fndFn) is a homogeneous linear recurrence sequence of order (n+2)(n+1)/2.
%C A379104 2nd esf of fndFn is given by A203245, with signature (4,-2,-6,4,2,-1).
%C A379104 3rd esf of fndFn is this sequence, with signature (7, -8, -27, 45, 24, -51, -3, 16, -1, -1).
%C A379104 4th esf of fndFn has signature (12,-28,-107,400,120,-1298,440,1408,-726,-560,296,83,-36,-4,1).
%C A379104 5th esf of fndFn has signature (20,-84,-423,3056,8,-25893,25956,73892,-105763,-77952,146160,30653,-86884,-3276,23499,-496,-2680,105,108,-4,-1).
%C A379104 The 6th esf of fndFn has signature (33,-240,-1671,22231,-12264,-477708,1054788,3271080,-10808292,-6412404,40815192,-4411686,-71500002,25737096,64629222,-28878366,-31047672,14128116,7759092,-3326280,-937860,364476,50568,-16577,-1143,264,9,-1).
%C A379104 The conjecture and signatures also apply to the first n distinct Lucas numbers (A000032).
%F A379104 a(n) = sum of F(i(1))*F(i(2))*F(i(3)) over all indices i(1)<i(2)<i(3)<=i(n).
%e A379104 a(2) = 1*2*3 + 1*2*5 + 1*3*5 + 2*3*5 = 61.
%t A379104 z = 50; w[i_] := Fibonacci[i];
%t A379104 t[n_] := Table[w[i], {i, 2, n}]
%t A379104 v[n_] := SymmetricPolynomial[3, t[n]]
%t A379104 tt = Table[v[n], {n, 4, 25}]
%Y A379104 Cf. A000032, A000045, A203245.
%K A379104 nonn
%O A379104 3,1
%A A379104 _Clark Kimberling_, Dec 16 2024
%E A379104 Deleted program and link based on a conjecture. - _N. J. A. Sloane_, Dec 22 2024