OFFSET
0,3
COMMENTS
This sequence is derived from a particular case of a general recurrence relation expressed by B(0) = x, B(1) = y and B(n) = n*B(n-1)/B(n-2), for n > 1 and {x,y} any pair of nonzero real numbers. Scatter plots of sequences of this kind exhibit a particular pattern that suggests the following conjecture:
lim_{n->infinity} B(6n+i)/(6n+i) = C_i and C_i != C_j for 0 < i < j < 7.
This means that B(n)/n approaches a cycle of six different constant values which depend on the particular chosen seed {x,y}. In this particular case the seed is {1,1} and the corresponding conjectured constant limits {C_1, C_2, C_3, C_4, C_5, C_6} are approximately {0.431, 0.615, 1.426, 2.319, 1.626, 0.701}. The corresponding constant limits for a generic seed {x,y} are respectively {C_1*y, C_2*y/x, C_3/x, C_4/y, C_5*x/y, C_6*x}. If x and y are not both positive then four of these constants are negative and two are positive.
FORMULA
a(n) = numerator(b(n)), where b(0) = b(1) = 1 and b(n) = n!/Product_{j=1..n-2} a(j), for n > 1.
MATHEMATICA
b[0]=1; b[1]=1;
b[n_]:=b[n]=n*b[n-1]/b[n-2]
(* Table[b[j], {j, 1, 2^10}]//ListPlot *)
Table[Numerator@b[j], {j, 0, 2^5}]
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Andres Cicuttin, Nov 18 2019
STATUS
approved