cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A357288 a(n) = (1/4)*A357287(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 5, 12, 23, 42, 85, 144, 251, 454, 753, 1244, 2183, 3506, 5725, 9736, 15539, 24926, 41993, 66228, 105823, 176138, 277173, 439648, 729099, 1141430, 1807713, 2981900, 4663991, 7361890, 12124173, 18918072, 29840739, 49020942
Offset: 0

Views

Author

Clark Kimberling, Oct 02 2022

Keywords

Crossrefs

Cf. A357287.

Programs

  • Mathematica
    s[n_] := s[n] = Select[Subsets[Range[n]], Length[#] >= 3 &];
    a[n_] := Select[s[n], #[[1]] + #[[2]] + #[[3]] == #[[-1]] &]
    (1/4)*Table[Length[a[n]], {n, 0, 15}]

Formula

a(n) = 2*a(n-1) + a(n-2) - 6*a(n-4) - 4*a(n-5) + 16*a(n-6) - 8*a(n-7).
G.f.: x^6/((-1 + x)^2 (-1 + 2 x^2) (-1 + 4 x^3)).

A357285 a(n) = number of subsets S of {1,2,...,n} having more than 2 elements such that (sum of least three elements of S) < max(S).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 8, 32, 104, 304, 792, 1920, 4520, 10192, 22392, 48416, 102856, 215664, 448792, 925632, 1897064, 3872016, 7868344, 15936096, 32208136, 64946096, 130738776, 262886656, 527990696, 1059498576, 2124829944, 4258791328, 8532044360, 17087943920
Offset: 0

Views

Author

Clark Kimberling, Oct 02 2022

Keywords

Examples

			The 8 relevant subsets of {1,2,3,4,5,6,7} are {1,2,3,7}, {1,2,3,4,7}, {1,2,3,5,7}, {1,2,3,6,7}, {1,2,3,4,5,7}, {1,2,3,4,6,7}, {1,2,3,5,6,7}, and {1,2,3,4,5,6,7}.
		

Crossrefs

Programs

  • Mathematica
    s[n_] := s[n] = Select[Subsets[Range[n]], Length[#] >= 3 &];
    a[n_] := Select[s[n], #[[1]] + #[[2]] + #[[3]] < #[[-1]] &]
    Table[Length[a[n]], {n, 0, 18}]

Formula

a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) - 6*a(n-4) + 8*a(n-5) + 24*a(n-6) - 40*a(n-7).
G.f.: -((8 x^7)/((-1 + x)^2 (-1 + 2 x) (-1 + 2 x^2) (-1 + 4 x^3))).

A357289 a(n) = number of subsets S of {1,2,...,n} having more than 2 elements such that (sum of least three elements of S) > max(S).

Original entry on oeis.org

0, 0, 0, 1, 5, 16, 38, 83, 167, 314, 572, 1021, 1757, 3004, 5082, 8439, 13971, 23086, 37576, 61281, 99833, 160912, 259878, 420283, 672847, 1081058, 1739124, 2774021, 4439701, 7121188, 11326386, 18087487, 28944587, 45962070, 73268704, 117090409, 185684721, 295697784, 472033278, 747983491
Offset: 0

Views

Author

Clark Kimberling, Oct 02 2022

Keywords

Examples

			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}.
		

Crossrefs

Programs

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

Formula

a(n) = 3*a(n-1) - a(n-2) - a(n-3) - 6*a(n-4) + 2*a(n-5) + 20*a(n-6) - 24*a(n-7) + 8*a(n-8).
G.f.: (x^3 (-1 - 2 x - 2 x^2 + 4 x^3 + 4 x^4))/((-1 + x)^3 (-1 + 2 x^2) (-1 + 4 x^3)).
Showing 1-3 of 3 results.