A145103 -id:A145103 - cp's OEIS frontend

A145102 a(0) = a(1) = 1. a(n+1) = floor(n*a(n)/a(n-1)), for n >= 1.

1, 1, 1, 2, 6, 12, 10, 5, 3, 4, 12, 30, 27, 10, 4, 5, 18, 57, 53, 16, 5, 6, 25, 91, 83, 21, 6, 7, 31, 124, 116, 28, 7, 8, 37, 157, 148, 33, 8, 9, 43, 191, 182, 40, 9, 9, 45, 230, 240, 50, 10, 10, 51, 265, 275, 56, 11, 11, 57, 300, 310, 62, 12, 12, 63, 336, 346, 67, 12, 12, 69, 402

Offset: 0

Leroy Quet, Oct 01 2008

At n = 62 the sequence settles down and becomes quasi-periodic with a 6-loop. - Ctibor O. Zizka, Feb 21 2025

Michel Marcus, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-5,6,-5,4,-3,2,-1,0,1,-2,3,-4,5,-6,5,-4,3,-2,1).

Cf. A145103.

{print1(a=1, ",", b=1, ","); for(n=2, 71, print1(c=floor((n-1)*b/a), ","); a=b; b=c)} \\ Klaus Brockhaus, Oct 02 2008

From Ctibor O. Zizka, Feb 21 2025: (Start)
For n >= 62,
a(n) = n/6 + floor((n - 2)*(n - 3)/12) - 1 if n mod 6 = 0,
a(n) = n if n mod 6 = 1,
a(n) = 12 if n mod 6 = 2 or 3,
a(n) = n - 1 if n mod 6 = 4,
a(n) = floor((n - 1)*(n - 2)/12) if n mod 6 = 5. (End)

More terms from Klaus Brockhaus and R. J. Mathar, Oct 02 2008

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.

Original entry on oeis.org

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.

A329813 a(n) = denominator(b(n)), where b(0) = b(1) = 1 and b(n) = n*b(n-1)/b(n-2) for n >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 5, 5, 7, 14, 6, 15, 275, 11, 91, 637, 14, 200, 935, 187, 247, 12103, 245, 22, 47311, 4301, 247, 112385, 5733, 2772, 1372019, 11339, 38285, 398164, 2184, 86394, 23324323, 56695, 21793, 69977323, 77064, 49368, 434680565, 464899, 937099, 6018049778, 635778, 2128995, 93977938153

Offset: 0

Andres Cicuttin, Nov 21 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. A329654 (numerators), 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[Denominator@b[j], {j, 0, 2^5}]

a(n) = denominator(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

A145102 a(0) = a(1) = 1. a(n+1) = floor(n*a(n)/a(n-1)), for n >= 1.

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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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A329813 a(n) = denominator(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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