A134067 -id:A134067 - cp's OEIS frontend

A288537 Array A(b,n) by upward antidiagonals (b>1, n>0): the eventual period of the RATS sequence in base b starting from n; 0 is for infinity.

1, 3, 1, 2, 3, 1, 2, 2, 3, 1, 8, 2, 2, 3, 1, 4, 8, 2, 2, 3, 1, 3, 4, 8, 2, 2, 3, 1, 2, 3, 2, 8, 2, 2, 3, 1, 0, 2, 3, 4, 2, 2, 2, 3, 1, 28, 0, 2, 3, 4, 8, 2, 2, 3, 1, 90, 28, 8, 2, 6, 2, 8, 2, 2, 3, 1, 8, 90, 28, 0, 2, 3, 4, 8, 2, 2, 3, 1, 72, 8, 90, 28, 0, 2

Offset: 2

Andrey Zabolotskiy, Jun 11 2017

Eventual period of n under the mapping x->A288535(b,x), or 0 if there is a divergence and thus no eventual period.
For b = 3*2^m - 2 with m>1, row b contains all sufficiently large even integers if m is odd, or just all sufficiently large integers if m is even.
For b = 1 or 10 (mod 18) or b = 1 (mod (2^q-1)^2) with q>2, there are 0's in row b.
Conway conjectured that in row (base) 10, all 0's correspond to the same divergent RATS sequence called the Creeper (A164338). In Thiel's terms, it is quasiperiodic with quasiperiod 2, i.e., after every 2 steps the number of one of the digits (in this case, 3 or 6) increases by 1 while other digits stay unchanged. In other bases, 0's may correspond to different divergent RATS sequences. Thiel conjectured that the divergent RATS sequences are always quasiperiodic.

			In base 3, the RATS mapping acts as 1 -> 2 -> 4 (11 in base 3) -> 8 (22 in base 3) -> 13 (112 in base 3) -> 4, which has already been seen 3 steps ago, so A(3,1)=3.
The array begins:
1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, ...
2, 2, 2, 2, 2, 2, ...
2, 2, 2, 2, 2, 2, ...
8, 8, 8, 8, 2, 8, ...
4, 4, 2, 4, 4, 2, ...
3, 3, 3, 3, 6, 3, ...
2, 2, 2, 2, 2, 2, ...
0, 0, 8, 0, 0, 8, ...
28, 28, 28, 28, 2, 28, ...
90, 90, 90, 90, 90, 90 ...

Curtis Cooper, RATS.
R. K. Guy, Conway's RATS and other reversals, Amer. Math. Monthly, 96 (1989), 425-428.
S. Shattuck and C. Cooper, Divergent RATS sequences, Fibonacci Quart., 39 (2001), 101-106.
J. Thiel, Conway’s RATS Sequences in Base 3, Journal of Integer Sequences, 15 (2012), Article 12.9.2.
J. Thiel, On RATS sequences in general bases, Integers, 14 (2014), #A50.
Eric Weisstein's World of Mathematics, RATS Sequence.
Index entries for sequences related to RATS: Reverse, Add Then Sort

Cf. A004000, A036839, A114611 (row 10), A161593, A288535, A288536 (column 1).

A(2^t,1)=t.

A(3,3^A134067(p)-1)=p+3.

A296965 Expansion of x(1 - x + 2x^2) / ((1 - x)(1 - 2x)).

Original entry on oeis.org

0, 1, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182

Offset: 0

J. Devillet, Dec 22 2017

a(n) = A000225(n)-1, a(0)=0, a(1)=1. Number of quasilinear weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1<...

Essentially the same as A095121 and A000918. - R. J. Mathar, Jan 02 2018

Colin Barker, Table of n, a(n) for n = 0..1000
J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017-2018.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A000918, A095121, A134067.

CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* or *)
LinearRecurrence[{3, -2}, {0, 1, 2, 6}, 34] (* Michael De Vlieger, Dec 22 2017 *)

concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).
a(n) = 2^n - 2 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3. (End)
a(n) = A134067(n-2) for n >= 3. - Georg Fischer, Oct 30 2018
E.g.f.: 1 + exp(x)*(exp(x) - 2) + x. - Stefano Spezia, May 07 2023

A134066 Triangle = A134060 + A124928 - A007318.

Original entry on oeis.org

1, 2, 4, 2, 8, 4, 2, 12, 12, 4, 2, 16, 24, 16, 4, 2, 20, 40, 40, 20, 4, 2, 24, 60, 80, 60, 24, 4, 2, 28, 84, 140, 140, 84, 28, 4, 2, 32, 112, 224, 280, 224, 112, 32, 42, 36, 144, 336, 504, 504, 336, 144, 36, 4

Offset: 0

Gary W. Adamson, Oct 06 2007

Row sums = A134067: (1, 6, 14, 30, 62, 126, ...).

			First few rows of the triangle:
  1;
  2,  4;
  2,  8,  4;
  2, 12, 12,  4;
  2, 16, 24, 16,  4;
  2, 20, 40, 40, 20,  4;
  ...

Cf. A134060, A124928, A134067.

Triangle as infinite lower triangular matrix, (A134060 + A124928 - A007318).

A135914 a(n) = 43^n - 22^n - 1.

Original entry on oeis.org

1, 7, 27, 91, 291, 907, 2787, 8491, 25731, 77707, 234147, 704491, 2117571, 6360907, 19099107, 57330091, 172055811, 516298507, 1549157667, 4647997291, 13945040451, 41837218507, 125515849827, 376555938091, 1129684591491, 3389087328907, 10167329095587

Offset: 0

N. J. A. Sloane, Mar 07 2008

G. S. Lueker, Some techniques for solving recurrences, Computing Surveys, 12 (1980), 419-436.

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A134067.

Table[4*3^n-2*2^n-1,{n,0,30}] (* or *) LinearRecurrence[{6,-11,6},{1,7,27},30] (* Harvey P. Dale, Aug 26 2019 *)

From Gary W. Adamson, Mar 08 2008: (Start)
Inverse binomial transform = A134067: (1, 6, 14, 30, 62, 126, ...).
Second inverse binomial transform = (1, 5, 3, 5, 3, 5, 3, 5, ...). (End)
From Colin Barker, Aug 13 2012: (Start)
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3).
G.f.: (1+x-4*x^2)/((1-x)*(1-2*x)*(1-3*x)). (End)

Showing 1-4 of 4 results.

cp's OEIS Frontend

A288537 Array A(b,n) by upward antidiagonals (b>1, n>0): the eventual period of the RATS sequence in base b starting from n; 0 is for infinity.

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A296965 Expansion of x(1 - x + 2x^2) / ((1 - x)(1 - 2x)).

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A134066 Triangle = A134060 + A124928 - A007318.

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A135914 a(n) = 43^n - 22^n - 1.

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cp's OEIS Frontend

A288537 Array A(b,n) by upward antidiagonals (b>1, n>0): the eventual period of the RATS sequence in base b starting from n; 0 is for infinity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A296965 Expansion of x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A134066 Triangle = A134060 + A124928 - A007318.

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Author

Keywords

Comments

Examples

Crossrefs

Formula

A135914 a(n) = 4*3^n - 2*2^n - 1.

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A296965 Expansion of x(1 - x + 2x^2) / ((1 - x)(1 - 2x)).

A135914 a(n) = 43^n - 22^n - 1.