A329813 -id:A329813 - cp's OEIS frontend

A329654 a(n) = numerator(b(n)), where b(0) = b(1) = 1 and b(n) = n*b(n-1)/b(n-2) for n >= 1.

1, 1, 2, 6, 12, 10, 5, 7, 28, 72, 180, 275, 55, 91, 2548, 252, 3600, 18700, 187, 1729, 12103, 5880, 13200, 473110, 4301, 247, 786695, 171990, 16632, 5488076, 124729, 38285, 27871480, 550368, 3110184, 23324323, 56695, 1416545, 559818584, 3236688, 2073456, 4781486215, 2324495, 937099, 12036099556

Offset: 0

Andres Cicuttin, Nov 18 2019

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.

Cf. A329813 (denominators), A145102, A145103.

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}]

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.

cp's OEIS Frontend

A329654 a(n) = numerator(b(n)), where b(0) = b(1) = 1 and b(n) = n*b(n-1)/b(n-2) for n >= 1.

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