A203299 -id:A203299 - cp's OEIS frontend

A203246 Second elementary symmetric function of the first n terms of (1,1,2,2,3,3,4,4,...).

1, 5, 13, 31, 58, 106, 170, 270, 395, 575, 791, 1085, 1428, 1876, 2388, 3036, 3765, 4665, 5665, 6875, 8206, 9790, 11518, 13546, 15743, 18291, 21035, 24185, 27560, 31400, 35496, 40120, 45033, 50541, 56373, 62871, 69730, 77330, 85330, 94150, 103411, 113575

Offset: 2

Clark Kimberling, Dec 31 2011

Second subdiagonal of A246117. - Peter Bala, Aug 15 2014

Sela Fried, On A203246, 2024.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A203298, A203299, A246117, A212523 (odd bisection), A103220 (even bisection).

f[k_] := Floor[(k + 1)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]  (* A203246 *)

Conjectural o.g.f.: x^2*(1 + 3*x + x^2 + x^3)/((1 - x^2)^3*(1 - x)^2). - Peter Bala, Aug 15 2014
Conjectural closed form: 64*a(n) = 2*n^2 -16*n/3 -3 +16*n^3/3 +2*n^4 +(-1)^n *(3-2*n^2). - R. J. Mathar, Oct 01 2016
Both conjectures are true. See link. - Sela Fried, Dec 22 2024

A203298 Second elementary symmetric function of the first n terms of (1,2,2,3,3,4,4,5,5...).

Original entry on oeis.org

2, 8, 23, 47, 91, 151, 246, 366, 540, 750, 1037, 1373, 1813, 2317, 2956, 3676, 4566, 5556, 6755, 8075, 9647, 11363, 13378, 15562, 18096, 20826, 23961, 27321, 31145, 35225, 39832, 44728, 50218, 56032, 62511, 69351, 76931, 84911, 93710, 102950

Offset: 2

Clark Kimberling, Dec 31 2011

nonn

Cf. A203246, A203299.

f[k_] := Floor[(k + 2)/2]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]  (* A203298 *)

Empirical g.f.: x^2*(2 + 4*x + 3*x^2 - 3*x^3 - x^4 + x^5) / ((1 - x)^5*(1 + x)^3). - Colin Barker, Aug 15 2014
Conjectures from Colin Barker, Jan 04 2018: (Start)
a(n) = (6*n^4 + 40*n^3 + 48*n^2 - 112*n) / 192 for n even.
a(n) = (6*n^4 + 40*n^3 + 36*n^2 - 136*n + 54) / 192 for n odd.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n>9.
(End)

cp's OEIS Frontend

A203246 Second elementary symmetric function of the first n terms of (1,1,2,2,3,3,4,4,...).

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