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Colin Barker, <a href="/A318111/b318111.txt">Table of n, a(n) for n = 1..1000</a>
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G.f.: x*(1 - 3*x + 3*x^2 + 6*x^3 - 14*x^4 + 11*x^5 - 3*x^6) / (1 - x)^4.
a(n) = n*(8 - 3*n + n^2) / 6 for n>3.
(PARI) Vec(x*(1 - 3*x + 3*x^2 + 6*x^3 - 14*x^4 + 11*x^5 - 3*x^6) / (1 - x)^4 + O(x^50)) \\ Colin Barker, Aug 31 2018
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a(n) = C(n,3) + n except for n = 2, 3 because all 1-intersecting families of 2-sets of size n > 3 can be interpreted as graphs with no independent edges. On n > 3 nodes, the only possibilities are triangles (C(n,3) possibilities) and stars (n possibilities, except for n=2,3).
This is because all 1-intersecting families of 2-sets of size n > 3 can be interpreted as graphs with no independent edges. On n > 3 nodes, the only possibilities are triangles (C(n,3) possibilities) and stars (n possibilities, except for n=2,3).
a(n) = C(n,3) + n except for n=2,3.
A318111 := n -> `if`(n<=3, 1, n*(8 - 3*n + n^2)/6):
seq(A318111(n), n=1..30); # Peter Luschny, Sep 05 2018
CoefficientList[Series[x*(1 - 3*x + 3*x^2 + 6*x^3 - 14*x^4 + 11*x^5 - 3*x^6) / (1 - x)^4, {x, 0, 50}], x] (* Stefano Spezia, Aug 31 2018 *)
a(n) = A000125(n-1) except for n = 2, 3.