A145102 -id:A145102 - cp's OEIS frontend

A145103 a(0) = a(1) = 1. a(n+1) = ceiling(n*a(n)/a(n-1)), for n >= 1.

1, 1, 1, 2, 6, 12, 10, 5, 4, 7, 16, 23, 16, 9, 8, 13, 25, 31, 22, 13, 12, 19, 34, 40, 28, 17, 16, 25, 43, 49, 34, 21, 20, 31, 52, 58, 40, 25, 24, 37, 61, 66, 45, 29, 28, 43, 70, 75, 51, 33, 32, 49, 79, 84, 57, 37, 36, 55, 88, 93, 63, 41, 40, 61, 97, 102, 69, 45, 44, 67, 106, 111

Offset: 0

Leroy Quet, Oct 01 2008

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,1,-1,1,-1,1,-1).

Cf. A145102.

A[0]:= 1: A[1]:= 1:
for n from 1 to 99 do A[n+1]:= ceil(n*A[n]/A[n-1]) od:
seq(A[i],i=0..100); # Robert Israel, Dec 06 2022

nxt[{n_,a_,b_}]:={n+1,b,Ceiling[((n+1)b)/a]}; Join[{1},Transpose[ NestList[ nxt,{0,1,1},80]][[3]]] (* Harvey P. Dale, Mar 04 2013 *)

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

a(n)=if(n<37, return([1, 1, 1, 2, 6, 12, 10, 5, 4, 7, 16, 23, 16, 9, 8, 13, 25, 31, 22, 13, 12, 19, 34, 40, 28, 17, 16, 25, 43, 49, 34, 21, 20, 31, 52, 58, 40][n+1])); my(r=n%6,k=n\6); if(r==0, 6*k+3, r==1, 4*k+1, r==2, 4*k, r==3, 6*k+1, r==4, 9*k+7, 9*k+12) \\ Charles R Greathouse IV, Dec 07 2022

From Robert Israel, Dec 06 2022: (Start)
a(6*k) = 6*k+3 for k >= 7.
a(6*k+1) = 4*k+1.
a(6*k+2) = 4*k for k >= 1.
a(6*k+3) = 6*k+1 for k >= 1.
a(6*k+4) = 9*k+7 for k >= 1.
a(6*k+5) = 9*k+12 for k >= 6.
(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.

A381331 a(1) = a(2) = 1; for n > 2, a(n) = floor((n - 2)*a(n - 1)/a(n - 2)) + GCD(n - 2, a(n - 2)).

Original entry on oeis.org

1, 1, 2, 5, 8, 7, 5, 5, 8, 13, 15, 12, 9, 21, 31, 27, 14, 9, 11, 31, 54, 35, 16, 11, 16, 35, 55, 41, 21, 15, 21, 57, 85, 48, 19, 15, 28, 70, 93, 52, 24, 22, 38, 74, 84, 51, 30, 28, 44, 79, 88, 56, 33, 34, 55, 89, 144, 91, 39, 25, 38, 96, 155, 102, 42, 28, 44, 105, 160, 104

Offset: 1

Ctibor O. Zizka, Feb 20 2025

At n = 499 the sequence settles down and becomes quasi-periodic with a 6-loop. Empiricaly 3 >= a(n + 1)/a(n) >= 1/3. The system is sensitive to the choice of initial terms [a(1),a(2)]. Only some values of initial terms results in a 6-loop like this sequence, the vast majority of initial terms show a "noisy quasiperiodic" like structures in the plot. Trials made for [a(1), a(2)] from [1, 1] to [100, 100] and for n up to 70000. May it be the sequence converges to a 6-loop for some large enough n, independent on the choice of initial terms ?

			a(1) = 1
a(2) = 1
a(3) = floor(1*1/1) + GCD(1,1) = 2
a(4) = floor(2*2/1) + GCD(2,1) = 5
a(5) = floor(3*5/2) + GCD(3,2) = 8
and so on.

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,0,0,0,0,0,0,1,-1,1,-1,1,-1).

Cf. A133058, A145102.

a[n_] := a[n] = If[n < 3, 1, Floor[(n-2)*a[n-1]/a[n-2]] + GCD[n-2, a[n-2]]]; Array[a, 70] (* Amiram Eldar, Feb 20 2025 *)

For n >= 499:
if n mod 6 = 0, a(n) = 2*n - 1 + 2*((n/2) mod 2).
if n mod 6 = 1, a(n) = n + 2.
if n mod 6 = 2, a(n) = (n + 2)/2.
if n mod 6 = 3, a(n) = (n - 1)/2.
if n mod 6 = 4, a(n) = n - 2 - (n/2) mod 2.
if n mod 6 = 5, a(n) = 2*n - 6 + 3*((n + 1)/2 mod 2).

cp's OEIS Frontend

A145103 a(0) = a(1) = 1. a(n+1) = ceiling(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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A381331 a(1) = a(2) = 1; for n > 2, a(n) = floor((n - 2)*a(n - 1)/a(n - 2)) + GCD(n - 2, a(n - 2)).

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