%I #30 Sep 06 2018 11:51:40

%S 1,1,1,8,15,26,42,64,93,130,176,232,299,378,470,576,697,834,988,1160,

%T 1351,1562,1794,2048,2325,2626,2952,3304,3683,4090,4526,4992,5489,

%U 6018,6580,7176,7807,8474,9178,9920,10701,11522,12384,13288,14235,15226

%N Number of maximal 1-intersecting families of 2-sets of [n] = {1,2,...,n}.

%C 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).

%H Colin Barker, <a href="/A318111/b318111.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F From _Colin Barker_, Aug 31 2018: (Start)

%F G.f.: x*(1 - 3*x + 3*x^2 + 6*x^3 - 14*x^4 + 11*x^5 - 3*x^6)/(1 - x)^4.

%F a(n) = n*(8 - 3*n + n^2)/6 for n>3.

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>7.

%F (End)

%p A318111 := n -> `if`(n<=3, 1, n*(8 - 3*n + n^2)/6):

%p seq(A318111(n), n=1..30); # _Peter Luschny_, Sep 05 2018

%t 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 *)

%o (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

%Y a(n) = A000125(n-1) except for n = 2,3.

%Y Cf. A318112.

%K nonn,easy

%O 1,4

%A _Manfred Scheucher_ and _Felix Schroeder_, Aug 17 2018