A198693 -id:A198693 - cp's OEIS frontend

A156760 5*4^n-1.

4, 19, 79, 319, 1279, 5119, 20479, 81919, 327679, 1310719, 5242879, 20971519, 83886079, 335544319, 1342177279, 5368709119, 21474836479, 85899345919, 343597383679, 1374389534719, 5497558138879, 21990232555519, 87960930222079, 351843720888319

Offset: 0

Paul Curtz, Feb 15 2009

Second column of the array A132207, or, if this array is flattened, a(n)=A132207(A007583(n)).

			Binary.......................................Decimal
100................................................4
10011.............................................19
1001111...........................................79
100111111........................................319
10011111111.....................................1279
1001111111111...................................5119
100111111111111................................20479
10011111111111111..............................81919
1001111111111111111...........................327679
100111111111111111111........................1310719
10011111111111111111111......................5242879
1001111111111111111111111...................20971519
100111111111111111111111111.................83886079
10011111111111111111111111111..............335544319
1001111111111111111111111111111...........1342177279
... - _Philippe Deléham_, Feb 23 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..500

[5*4^n-1: n in [0..30]]; // Vincenzo Librandi, Jul 02 2011

Table[5*4^n - 1, {n, 0, 18}] (* Michael De Vlieger, Apr 20 2015 *)

a(n) mod 9 = A070403(n+2).
a(n+1) = 10*A083420(n)+9 .
a(n) = 5*A000302(n)-1.
a(n) = ( A024036(n+1)+A140529(n) )/2.
a(n) = 4a(n-1)+3, a(0)=4.
a(n) = A003947(n+1)-1 = 5*a(n-1)-4*a(n-2). G.f.: (4-x)/((1-x)(1-4x)). - R. J. Mathar, Feb 23 2009
a(n) = A198693(n) + 2^(2n+1). - Bob Selcoe, Apr 20 2015

Edited and extended by R. J. Mathar, Feb 23 2009

A259663 Square array T(n,k) read by antidiagonals upwards: "Dropping Times" in reduced Collatz sequences. (See "Comments" for definitions and explanation.)

Original entry on oeis.org

1, 13, 11, 5, 19, 7, 53, 3, 55, 47, 21, 35, 87, 79, 31, 213, 99, 23, 143, 223, 191, 85, 483, 407, 15, 95, 319, 127, 853, 739, 663, 271, 351, 63, 895, 767, 341, 1251, 1175, 1807, 863, 1599, 1407, 1279, 511

Offset: 2

Bob Selcoe, Jul 02 2015

Array starts T(2,1).
Coefficients are least residues in congruence classes T(n,k) mod 2^(n+k). Let T"(n,k) be all members of that class.
Let reduced Collatz sequences (R) be Collatz sequences (C) showing only odd terms; and let S be the initial term in R and C, denoted as R(S) and C(S), respectively.
Define "Dropping Time" (D(S)) as the first term in R(S) < its preimage (P(S)). For example, S=159: R(159) starts [159, 239, 359, 539, 809, 607, ...]; therefore, D(159) = 607 and P(159) = 809.
When S is in T"(n,k): k is the number of terms in R(S) from S to D(S), and n is the number of halving steps in C(S) from P(S) to D(S). So for S=159: since 159 == 31 mod 128 and T(2,5) = 31, there are 5 steps from S=159 to D(159) = 607 (239, 359, 539, 809, 607) and 2 halving steps from P(159) = 809 to D(159), i.e., 809*3+1 = 2428; 2428/2 = 1214, 1214/2 = 607.
Generally, when any term in R is in T"(n,k) k>=2, trajectories follow T"(n,k-i) {i=1..k-1}. At T"(n,1), the next term's congruence class is not clearly predictable. So for example S=159: 159 == 31 mod 128 (in T"(2,5)), 239 == 47 mod 64 (in T"(2,4)), 359 == 7 mod 32 (in T"(2,3)), 539 == 11 mod 16 (in T"(2,2)) and 809 == 1 mod 8 (in T"(2,1)). Since 607 == 95 mod 512 (in T"(4,5)), we know the subsequent term will be in T"(4,4), followed by terms in T"(4,3), T"(4,2) and T"(4,1).
Let T"(k) be members of all T"(n,k) holding k constant. Then T"(k) == (2^k-1) mod 2^(k+1). However, since S is the only possible term in R(S) that could be congruent to 0 mod 3, it makes sense to consider only terms congruent to {1,2} mod 3 when evaluating T"(k). Therefore, after the initial term in R(S), all subsequent terms in T"(k) are congruent to either:
i. {(2^k - 1), (6*4^((k-1)/2) - 1)} mod 3*2^(k+1) when k is odd; or
ii. {(3*4^(k/2) - 1), (5*4^(k/2) - 1)} mod 3*2^(k+1) when k is even.
The array yields a wide variety of interesting patterns and sub-patterns associated with the residues and quotients of the congruence classes. Perhaps analysis of these patterns could shed light on the nature of Collatz sequences, including the Collatz conjecture (i.e., all Collatz sequences terminate at 1).
From Bob Selcoe, Sep 30 2019: (Start)
From equations i and ii above, terms in T"(k) can be described as follows:
ia. for odd k: {T(2,k), T(m+3,k)} mod 3*2^(k+1) when k == 2^m - 1 mod 2^(m+1), m >= 1; or
iia. for even k: {T(2,k), T(3,k)} mod 3*2^(k+1).
(End)

			Array starts T(2,1):
n\k   1    2     3     4      5      6      7      8      9 ...
2:    1    11    7     47     31     191    127    767    511
3:    13   19    55    79     223    319    895    1279   3583
4:    5    3     87    143    95     63     1407   2303   1535
5:    53   35    23    15     351    1599   2431   4351   13823
6:    21   99    407   271    863    575    383    255    22015
7:    213  483   663   1807   3935   2623   12671  8447   5631
8:    85   739   1175  783    5983   14911  20863  57599  38399
9:    853  1251  2199  6927   10079  6719   4479   90367  235007
10:   341  227   151   11023  18271  55871  37247  24831  366079
For n >= 4: e.g., n=4, so j == 2 (mod 4). Select j=6, i=2 to find T(4,4). T(4,6) = 2^6 - 1 = 63. 2^(6-2)*3^2 - 1 mod 2^(4+6-2) = 143 mod 256 = T(4,4) = 143. Now instead select j=10, i=6 to find T(4,4). T(4,10) = 2^10 - 1 = 1023. 2^(10-6)*3^6 - 1 mod 2^(4+10-6) = 11663 mod 256 = 143. - _Bob Selcoe_, Jul 15 2017

Cf. A083420, A156760, A198693, A206372.

T(n, k) = if (n==2, if (k%2, 2^k-1, 3*2^k-1), if (n==3, if (k%2, 7*2^k-1, 5*2^k-1), mj = 2^(n-3) % 2^(n-2); mk = k % 2^(n-2); (2^k*3^(mj-mk) - 1) % 2^(n+k)));
tabl(nn) = matrix(nn, nn, n, k, T(n+1,k)); \\ Michel Marcus, Jul 10 2018

From Bob Selcoe, Jul 15 2017: (Start)
The array is constructed by the following:
T(2,k) = 2^k-1 when k is odd, T(2,k) = 3*2^k-1 when k is even; i.e., A083420((k-1)/2) and A198693(k/2) interleaved.
T(3,k) = 7*2^k-1 when k is odd; T(3,k) = 5*2^k-1 when k is even; i.e., A206372((k-1)/2) and A156760(k/2) interleaved.
For n >= 4: T(n,j) = 2^j-1, j == 2^(n-3) (mod 2^(n-2)); T(n,j-i) = least residue of 2^(j-i)*3^i - 1 mod 2^(n+j-i), 1 <= i < j. (See Example.)
(End)

A267614 Decimal representation of the n-th iteration of the "Rule 185" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 1, 11, 47, 191, 767, 3071, 12287, 49151, 196607, 786431, 3145727, 12582911, 50331647, 201326591, 805306367, 3221225471, 12884901887, 51539607551, 206158430207, 824633720831, 3298534883327, 13194139533311, 52776558133247, 211106232532991, 844424930131967

Offset: 0

Robert Price, Jan 18 2016

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Robert Price, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata

Cf. A198693, A267612, A267613.

rule=185; rows=20; ca=CellularAutomaton[rule,{{1},0},rows-1,{All,All}]; (* Start with single black cell *) catri=Table[Take[ca[[k]],{rows-k+1,rows+k-1}],{k,1,rows}]; (* Truncated list of each row *) Table[FromDigits[catri[[k]],2],{k,1,rows}]   (* Decimal Representation of Rows *)

Conjectures from Colin Barker, Jan 18 2016 and Apr 20 2019: (Start)
a(n) = 5*a(n-1) - 4*a(n-2) for n>3.
G.f.: (1-4*x+10*x^2-4*x^3) / ((1-x)*(1-4*x)).
(End)
Empirical a(n) = 3*4^(n-1)-1 = A198693(n-1) for n>1. - Colin Barker, Nov 25 2016 and Apr 20 2019

Removed an unjustified claim that Colin Barker's conjectures are correct. Removed a program based on a conjecture. - Michael De Vlieger, Jun 13 2022

A181358 Number of twiddle factors in the first stage of a Pease Radix 4 Fast Fourier Transform.

Original entry on oeis.org

8, 44, 188, 764, 3068, 12284, 49148, 196604, 786428, 3145724, 12582908, 50331644, 201326588, 805306364, 3221225468, 12884901884, 51539607548, 206158430204, 824633720828, 3298534883324, 13194139533308, 52776558133244, 211106232532988, 844424930131964

Offset: 0

Robert Koutsoyannis (bob.koutsoyannis(AT)gmail.com), Oct 14 2010

Number of twiddle factors in the first stage of a Pease Radix 4 Fast Fourier Transform.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Twiddle factor
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Vec(4*(2+x) / ((1-x) * (1-4*x)) + O(x^30)) \\ Colin Barker, Nov 26 2016

a(n) = 4*(3*4^n - 1).
From Colin Barker, Nov 26 2016: (Start)
a(n) = 4*A198693(n).
a(n) = 5*a(n-1) - 4*a(n-2) for n>1.
G.f.: 4*(2+x) / ((1-x) * (1-4*x)).
(End)

A281500 Reduced denominators of f(n) = (n+1)/(2^(2+n)-2) with A026741(n+1) as numerators.

Original entry on oeis.org

2, 3, 14, 15, 62, 63, 254, 255, 1022, 1023, 4094, 4095, 16382, 16383, 65534, 65535, 262142, 262143, 1048574, 1048575, 4194302, 4194303, 16777214, 16777215, 67108862, 67108863, 268435454, 268435455, 1073741822, 1073741823, 4294967294, 4294967295, 17179869182, 17179869183

Offset: 0

Paul Curtz, Jan 23 2017

f(n) = (n+1)/A000918(n+2) = 1/2, 2/6, 3/14, 4/30, 5/62, 6/126, 7/254, 8/510, 9/1022, 10/2046, 11/4094, 12/8190, ... .
Partial reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 3/63, 7/254, 4/255, 9/1022, 5/1023, 11/4094, 6/4095, ... = A026741(n+1)/a(n).
Full reduction: 1/2, 1/3, 3/14, 2/15, 5/62, 1/21, 7/254, ... = A111701(n+1)/(2, 3, 14, 15, 62, 21, ... )
A164555(n+1)/A027642(n) = 1/2, 1/6, 0, -1/30, 0, 1/42, ... = f(n) * A198631(n)/A006519(n+1) = 1, 1/2, 0, -1/4, 0, 1/2, ... .).
Via f(n), we go from the second fractional Euler numbers to the second Bernoulli numbers.
a(n) mod 10: periodic sequence of length 4: repeat [2, 3, 4, 5].
a(n) differences table:
.  2,   3, 14,  15,  62,   63, 254,  255, ...
.  1,  11,  1,  47,   1,  191,   1,  767, ... see A198693
. 10, -10, 46, -46, 190, -190, 766, -766, ... see A096045, from Bernoulli(2n).
Extension of a(n): a(-2) = -1, a(-1) = 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-4).

Cf. A000027, A000918, A001477, A006519, A026741, A027642, A096045, A111701, A117856, A164555, A198631, A198693, A209308, A267921.

a[n_] := (3+(-1)^n)*(2^(n+1)-1)/2; (* or *) a[n_] := If[EvenQ[n], 4^(n/2+1)-2, 4^((n+1)/2)-1]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Jan 24 2017 *)

Vec((2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)) + O(x^50)) \\ Colin Barker, Jan 24 2017

From Colin Barker, Jan 24 2017: (Start)
G.f.: (2 + 3*x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 5*a(n-2) - 4*a(n-4) for n>3. (End)
From Jean-François Alcover, Jan 24 2017: (Start)
a(n) = (3 + (-1)^n)*(2^(n + 1) - 1)/2.
a(n) = 4^((n + 1 + ((n + 1) mod 2))/2) - 1 - ((n + 1) mod 2). (End)
a(n) = a(n-2) + A117856(n+1) for n>1.
a(2*k) = 4^(k + 1) - 2, a(2*k+1) = a(2*k) + 1 = 4^(k+1) - 1.
a(2*k) + a(2*k+1) = A267921(k+1).

A303611 a(n) = (-1 - (-2)^(n-2)) mod 2^n.

Original entry on oeis.org

2, 1, 11, 7, 47, 31, 191, 127, 767, 511, 3071, 2047, 12287, 8191, 49151, 32767, 196607, 131071, 786431, 524287, 3145727, 2097151, 12582911, 8388607, 50331647, 33554431, 201326591, 134217727, 805306367, 536870911, 3221225471, 2147483647, 12884901887, 8589934591

Offset: 2

Vincenzo Librandi, May 07 2018

A198693 and A083420 interleaved. From 11 onwards, apparently A283651 and A290195 contain the same terms. - Bruno Berselli, May 07 2018

Olivier Rozier, Parity sequences of the 3x+1 map on the 2-adic integers and Euclidean embedding, arXiv:1805.00133 [math.DS], 2018-2025, see Definition 4.4 on p. 21.
Index entries for linear recurrences with constant coefficients, signature (1,4,-4).

All terms belong to A052955 and A180516.

Cf. A083420, A198693, A283651, A290195.

[IsOdd(n) select 2^(n-2)-1 else 3*2^(n-2)-1: n in [2..40]];

I:=[2,1,11]; [n le 3 select I[n] else Self(n-1)+4*Self(n-2)-4*Self(n-3): n in [1..35]];

Table[If[OddQ[n], 2^(n - 2) - 1, 3 2^(n - 2) - 1], {n, 2, 80}]
LinearRecurrence[{1, 4, -4}, {2, 1, 11}, 30]

a(n) = if (n%2, 2^(n-2) - 1, 3*2^(n-2) - 1); \\ Michel Marcus, May 30 2018

a(n) = 2^(n-2) - 1 for odd n, otherwise a(n) = 3*2^(n-2) - 1, with n>1.
From Bruno Berselli, May 07 2018: (Start)
O.g.f.: x^2*(2 - x + 2*x^2)/((1 - x)*(1 - 2*x)*(1 + 2*x)).
E.g.f.: (1 + 2*x - 4*exp(x) + exp(-2*x) + 2*exp(2*x))/4.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3).
a(n) = (2 + (-1)^n)*2^(n-2) - 1. (End)

cp's OEIS Frontend

A156760 5*4^n-1.

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A259663 Square array T(n,k) read by antidiagonals upwards: "Dropping Times" in reduced Collatz sequences. (See "Comments" for definitions and explanation.)

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A267614 Decimal representation of the n-th iteration of the "Rule 185" elementary cellular automaton starting with a single ON (black) cell.

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A181358 Number of twiddle factors in the first stage of a Pease Radix 4 Fast Fourier Transform.

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A281500 Reduced denominators of f(n) = (n+1)/(2^(2+n)-2) with A026741(n+1) as numerators.

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A303611 a(n) = (-1 - (-2)^(n-2)) mod 2^n.

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