A357291 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, 0, 0, 0, 1, 3, 8, 19, 42, 89, 185, 378, 766, 1544, 3102, 6220, 12459, 24939, 49902, 99831, 199692, 399417, 798871, 1597782, 3195608, 6391264, 12782580, 25565216, 51130493, 102261051, 204522172, 409044419, 818088918, 1636177921, 3272355933
Offset: 0
Examples
The 3 relevant subsets of {1,2,3,4,5,6,7} are {1, 2, 6}, {1, 2, 7}, {1, 2, 3, 7}.
Links
- Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,-1,1,3,-2).
Programs
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Mathematica
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}]
Formula
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^6/((-1 + x)^3 (1 + x) (-1 + 2 x) (1 + x + x^2)).
a(n) ~ A021025*2^n. - Stefano Spezia, Oct 03 2022
a(n) = 2^n/21 - n^2/12 + n/6 + O(1). Conjecture: a(n) = round(2^n/21 - n^2/12 + n/6). - Charles R Greathouse IV, Oct 11 2022
Comments