OFFSET
1,4
COMMENTS
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).
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
From Colin Barker, Aug 31 2018: (Start)
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.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>7.
(End)
MAPLE
A318111 := n -> `if`(n<=3, 1, n*(8 - 3*n + n^2)/6):
seq(A318111(n), n=1..30); # Peter Luschny, Sep 05 2018
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Manfred Scheucher and Felix Schroeder, Aug 17 2018
STATUS
approved