OFFSET
3,1
COMMENTS
From Andrew Howroyd, Aug 11 2017: (Start)
For n > 3 the irredundant sets are:
- 2*(2^n-1): any number of vertices from one side of the graph
- n^2: any two vertices on opposite sides
- n*(n-1)*(n-2): two vertices on one side and one non-opposing on the other
- n*(n-1)*(n-2)*(n-3)/4: two non-opposing vertices from each side
- 1: the empty set
(End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 3..1000
Eric Weisstein's World of Mathematics, Crown Graph
Eric Weisstein's World of Mathematics, Irredundant Set
Index entries for linear recurrences with constant coefficients, signature (7, -20, 30, -25, 11, -2).
FORMULA
a(n) = 2^(n+1) + n*(n^3 - 2*n^2 + 3*n + 2)/4 - 1 for n > 3. - Andrew Howroyd, Aug 11 2017
a(n) = 7*a(n-1) - 20*a(n-2) + 30*a(n-3) - 25*a(n-4) + 11*a(n-5) - 2*a(n-6) for n > 9.
G.f.: (x^3 (24 - 91 x + 119 x^2 - 53 x^3 - 37 x^4 + 44 x^5 - 12 x^6))/((-1 + x)^5 (-1 + 2 x)).
MATHEMATICA
Table[If[n == 3, 24, 2^(n + 1) + n*(n^3 - 2 n^2 + 3 n + 2)/4 - 1], {n, 3, 20}]
Join[{24}, LinearRecurrence[{7, -20, 30, -25, 11, -2}, {77, 178, 373, 724, 1331, 2364}, 20]]
CoefficientList[Series[(24 - 91 x + 119 x^2 - 53 x^3 - 37 x^4 + 44 x^5 - 12 x^6)/((-1 + x)^5 (-1 + 2 x)), {x, 0, 20}], x]
PROG
(PARI) a(n)=if(n<4, [4, 9, 24][n], 2^(n+1) + n*(n^3 - 2*n^2 + 3*n + 2)/4 - 1); \\ Andrew Howroyd, Aug 11 2017
(Magma) [24] cat [2^(n+1)+n*(n^3-2*n^2+3*n+2)/4-1: n in [4..40]]; // Vincenzo Librandi, Mar 17 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Aug 09 2017
EXTENSIONS
a(13)-a(32) from Andrew Howroyd, Aug 11 2017
STATUS
approved