A357291 -id:A357291 - cp's OEIS frontend

A357290 a(n) = number of subsets S of {1,2,...,n} having more than 2 elements such that (sum of least two elements of S) > difference between greatest two elements of S.

0, 0, 0, 1, 5, 15, 39, 91, 200, 424, 879, 1796, 3639, 7334, 14734, 29545, 59179, 118459, 237033, 474195, 948534, 1897228, 3794633, 7589460, 15179133, 30358498, 60717248, 121434769, 242869833, 485739983, 971480307, 1942960979, 3885922348, 7771845112

Offset: 0

Clark Kimberling, Oct 02 2022

			The 5 relevant subsets of {1,2,3,4} are {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, and {1, 2, 3, 4}.

Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,-1,1,3,-2).

Cf. A357291, A357292.

s[n_] := s[n] = Select[Subsets[Range[n]], Length[#] >= 3 &];
a[n_] := Select[s[n], #[[1]] + #[[2]] > #[[-1]] - #[[-2]] &]
Table[Length[a[n]], {n, 0, 15}]

a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) + 3*a(n-6) - 2*a(n-7).

G.f.: (x^3 (1 + 2 x + x^2 + x^3))/((-1 + x)^3 (1 + x) (-1 + 2 x) (1 + x + x^2)).

A357292 a(n) = number of subsets S of {1,2,...,n} having more than 2 elements such that (sum of least two elements of S) = difference between greatest two elements of S.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 11, 23, 47, 96, 193, 388, 778, 1558, 3118, 6239, 12480, 24963, 49929, 99861, 199725, 399454, 798911, 1597826, 3195656, 6391316, 12782636, 25565277, 51130558, 102261121, 204522247, 409044499, 818089003, 1636178012, 3272356029

Offset: 0

Clark Kimberling, Oct 02 2022

			The 2 relevant subsets of {1,2,3,4,5,6} are {1, 2, 5} and {1,2,3,6}.

Index entries for linear recurrences with constant coefficients, signature (2,1,-1,-2,-1,2).

Cf. A357290, A357291.

s[n_] := s[n] = Select[Subsets[Range[n]], Length[#] >= 3 &];
a[n_] := Select[s[n], #[[1]] + #[[2]] == #[[-1]] - #[[-2]] &]
Table[Length[a[n]], {n, 0, 16}]

a(n) = 2*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6).

G.f.: -(x^5/((-1 + x)^2 (1 + x) (-1 + 2 x) (1 + x + x^2))).

cp's OEIS Frontend

A357290 a(n) = number of subsets S of {1,2,...,n} having more than 2 elements such that (sum of least two elements of S) > difference between greatest two elements of S.

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