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

6, 61, 389, 2066, 9962, 45594, 202344, 881859, 3801171, 16275292, 69399116, 295177196, 1253532482, 5318285553, 22550198601, 95580699774, 405034367814, 1716140731030, 7270703692340, 30801852323495, 130485697292231, 552764498063256, 2341595675572344

Offset

3,1

Comments

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.

2nd esf of fndFn is given by A203245, with signature (4,-2,-6,4,2,-1).

3rd esf of fndFn is this sequence, with signature (7, -8, -27, 45, 24, -51, -3, 16, -1, -1).

4th esf of fndFn has signature (12,-28,-107,400,120,-1298,440,1408,-726,-560,296,83,-36,-4,1).

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).

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).

The conjecture and signatures also apply to the first n distinct Lucas numbers (A000032).

Links

Table of n, a(n) for n=3..25.

Formula

a(n) = sum of F(i(1))*F(i(2))*F(i(3)) over all indices i(1)<i(2)<i(3)<=i(n).

Example

a(2) = 1*2*3 + 1*2*5 + 1*3*5 + 2*3*5 = 61.

Mathematica

z = 50; w[i_] := Fibonacci[i];
t[n_] := Table[w[i], {i, 2, n}]
v[n_] := SymmetricPolynomial[3, t[n]]
tt = Table[v[n], {n, 4, 25}]

Crossrefs

Cf. A000032, A000045, A203245.

Sequence in context: A326739 A055331 A024523 * A252873 A068171 A069633

Adjacent sequences: A379101 A379102 A379103 * A379105 A379106 A379107

Keyword

nonn

Author

Clark Kimberling, Dec 16 2024

Extensions

Deleted program and link based on a conjecture. - N. J. A. Sloane, Dec 22 2024

Status

approved