A329654 - OEIS

%I #35 Dec 12 2019 11:39:55

%S 1,1,2,6,12,10,5,7,28,72,180,275,55,91,2548,252,3600,18700,187,1729,

%T 12103,5880,13200,473110,4301,247,786695,171990,16632,5488076,124729,

%U 38285,27871480,550368,3110184,23324323,56695,1416545,559818584,3236688,2073456,4781486215,2324495,937099,12036099556

%N a(n) = numerator(b(n)), where b(0) = b(1) = 1 and b(n) = n*b(n-1)/b(n-2) for n >= 1.

%C 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:

%C lim_{n->infinity} B(6n+i)/(6n+i) = C_i and C_i != C_j for 0 < i < j < 7.

%C 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.

%F 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.

%t b[0]=1; b[1]=1;

%t b[n_]:=b[n]=n*b[n-1]/b[n-2]

%t (* Table[b[j],{j,1,2^10}]//ListPlot *)

%t Table[Numerator@b[j], {j, 0, 2^5}]

%Y Cf. A329813 (denominators), A145102, A145103.

%K nonn,frac

%O 0,3

%A _Andres Cicuttin_, Nov 18 2019