A357282 a(n) = number of subsets S of {1,2,...,n} having more than 1 element such that (difference between least two elements of S) = difference between greatest two elements of S.

0, 0, 1, 4, 9, 18, 33, 60, 109, 202, 381, 732, 1425, 2802, 5545, 11020, 21957, 43818, 87525, 174924, 349705, 699250, 1398321, 2796444, 5592669, 11185098, 22369933, 44739580, 89478849, 178957362, 357914361, 715828332, 1431656245, 2863312042, 5726623605

Offset: 0

Author

Clark Kimberling, Sep 27 2022

Keywords

• nonn
• easy

Examples

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

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,5,-2)

Crossrefs

Cf. A000295, A272144.

Programs

Mathematica

s[n_] := s[n] = Select[Subsets[Range[n]], Length[#] >= 2 &];
a[n_] := Select[s[n], #[[2]] - #[[1]] == #[[-1]] - #[[-2]] &];
Table[Length[a[n]], {n, 0, 18}]

Formula

a(n) = A000295(n)-2*A272144(n-2) for n >= 2.

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

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