OFFSET
2,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Chordless Cycle
Eric Weisstein's World of Mathematics, Johnson Graph
Eric Weisstein's World of Mathematics, Triangular Graph
FORMULA
a(n) = Sum_{k=4..n} n!/(2*k*(n-k)!). - Andrew Howroyd, Jan 04 2018
a(n) = n*((3 - 2*n)*n + 6*Hypergeometric3F1[1, 1, 1 - n; 2; -1] - 7)/12. - Eric W. Weisstein, Jan 05 2018
-(-1 + n)*n*(1 + n) + (4 + 2*(-1 + n))*a(n) + (-6 - 2*(-1 + n))*a(n + 1) + 2*a(n + 2) = 0. - Eric W. Weisstein, Jan 07 2018
EXAMPLE
From Andrew Howroyd, Jan 04 2018: (Start)
Vertices can be represented by a pair of integers with 12 being the same as 21.
a(4) = 3 because the possible cycles are: -12-23-34-41-, -12-24-43-31-, -13-32-24-41-.
a(5) = 27 because there are 15 cycles of length 4 and 12 cycles of length 5.
(End)
MAPLE
A297670List := proc(n) local A, R, f, i; A:=[0, 0, 0, 6, 54, 354, 2274]; R:=NULL;
f := i -> (24*(12*A[1]-33*A[2]+23*A[3]+3*A[4]-5*A[5])-(4*(90*A[1]-255*A[2]
+212*A[3]-26*A[4]-31*A[5]+16*A[6])+(-208*A[1]+618*A[2]-604*A[3]+197*A[4]
+15*A[5]-35*A[6]+(82*A[1]-257*A[2]+285*A[3]-137*A[4]+27*A[5]+6*A[6]+
(-20*A[1]+66*A[2]-83*A[3]+52*A[4]-18*A[5]+2*A[6]+(+2*A[1]-7*A[2]+10*A[3]
-8*A[4]+4*A[5]-A[6])*i)*i)*i)*i)*i)/((-24+(17+(i-6)*i)*i)*i);
for i from 1 to n do if i<7 then R:=R, A[i+1]/2 else A[1]:=A[2]; A[2]:=A[3];
A[3]:=A[4]; A[4]:=A[5]; A[5]:=A[6]; A[6]:=A[7]; A[7]:=f(i); R:=R, A[7]/2 fi od;
R end: A297670List(22); # Peter Luschny, Jan 06 2018
MATHEMATICA
Table[Sum[n!/(2 k (n - k)!), {k, 4, n}], {n, 2, 20}]
Table[n ((3 - 2 n) n + 6 HypergeometricPFQ[{1, 1, 1 - n}, {2}, -1] - 7)/12, {n, 2, 20}]
RecurrenceTable[{-(-1 + n) n (1 + n) + (4 + 2 (-1 + n)) a[n] + (-6 - 2 (-1 + n)) a[n + 1] + 2 a[n + 2] == 0, a[1] == 0, a[2] == 0}, a[n], {n, 2, 20}]
PROG
(PARI) a(n)={sum(k=4, n, n!/(2*k*(n-k)!))} \\ Andrew Howroyd, Jan 04 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Eric W. Weisstein, Jan 02 2018
EXTENSIONS
Terms a(8) and beyond from Andrew Howroyd, Jan 04 2018
STATUS
approved