A086224 -id:A086224 - cp's OEIS frontend

A198276 a(n) = 19*2^n-1.

18, 37, 75, 151, 303, 607, 1215, 2431, 4863, 9727, 19455, 38911, 77823, 155647, 311295, 622591, 1245183, 2490367, 4980735, 9961471, 19922943, 39845887, 79691775, 159383551, 318767103, 637534207, 1275068415, 2550136831, 5100273663, 10200547327, 20401094655

Offset: 0

Jeremy Gardiner, Oct 23 2011

			a(2) = 19*2^2-1 = 75.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A086224, A110288.

for j = 0 to 30 : print str$((19*2^j)-1)+", "; : next j

[19*2^n-1: n in [0..30]]; // Vincenzo Librandi, Oct 28 2011

A198276:=n->19*2^n-1; seq(A198276(n), n=0..30); # Wesley Ivan Hurt, Jun 11 2014

19*2^Range[0,30]-1 (* or *) LinearRecurrence[{3,-2},{18,37},40] (* Harvey P. Dale, Dec 18 2013 *)

a(n+1) = 2*a(n) + 1.
G.f.: ( 18-17*x ) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Oct 25 2011
a(n) + a(n-1)^2 = (a(n-1)+1)^2. - Vincenzo Librandi, Jun 11 2014

A171389 a(n) = 21*2^n - 1.

Original entry on oeis.org

20, 41, 83, 167, 335, 671, 1343, 2687, 5375, 10751, 21503, 43007, 86015, 172031, 344063, 688127, 1376255, 2752511, 5505023, 11010047, 22020095, 44040191, 88080383, 176160767, 352321535, 704643071, 1409286143, 2818572287

Offset: 0

Vincenzo Librandi, Dec 07 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A086224, A175805.

for j = 0 to 30 : print str$((21*2^j)-1)+", "; : next j [Jeremy Gardiner, Oct 23 2011]

I:=[20, 41]; [n le 2 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jul 06 2012

A171389:=n->21*2^n-1; seq(A171389(n), n=0..40); # Wesley Ivan Hurt, Jun 11 2014

CoefficientList[Series[(20-19*x)/((1-x)*(1-2*x)),{x,0,40}],x] (* Vincenzo Librandi, Jul 06 2012 *)

a(n+1) = 2*a(n) + 1.
G.f.: (20-19*x)/((1-x)*(1-2*x)). - Vincenzo Librandi, Jul 06 2012
a(n) = 3*a(n-1) - 2*a(n-2). - Vincenzo Librandi, Jul 06 2012
a(n) + a(n-1)^2 = (a(n-1) + 1)^2. - Vincenzo Librandi, Jun 11 2014

Edited by Jon E. Schoenfield, Jun 23 2010

Offset changed to 0 and first formula corrected by Jeremy Gardiner, Oct 23 2011

A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 15, 8, 11, 6, 31, 16, 23, 12, 9, 63, 32, 47, 24, 19, 10, 127, 64, 95, 48, 39, 20, 13, 255, 128, 191, 96, 79, 40, 27, 14, 511, 256, 383, 192, 159, 80, 55, 28, 17, 1023, 512, 767, 384, 319, 160, 111, 56, 35, 18, 2047, 1024, 1535, 768, 639

Offset: 1

Clark Kimberling, Jun 11 2011

Row 1:  A000225 (-1+2^n)
Row 2:  A000079 (2^n)
Row 3:  A055010
Row 4:  3*A000079
Row 5:  A153894
Row 6:  5*A000079
Row 7:  A086224
Row 8:  A005009
Row 9:  A052996
For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The six sequences and dispersions are listed here:
...
A191663=dispersion of A042948 (0 or 1 mod 4 and >1)
A054582=dispersion of A005843 (0 or 2 mod 4 and >1; evens)
A191664=dispersion of A014601 (0 or 3 mod 4 and >1)
A191665=dispersion of A042963 (1 or 2 mod 4 and >1)
A191448=dispersion of A005408 (1 or 3 mod 4 and >1, odds)
A191666=dispersion of A042964 (2 or 3 mod 4)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191663 has 1st col A042964, all else A042948
A054582 has 1st col A005408, all else A005843
A191664 has 1st col A042963, all else A014601
A191665 has 1st col A014601, all else A042963
A191448 has 1st col A005843, all else A005408
A191666 has 1st col A042948, all else A042964
...
There is a formula for sequences of the type "(a or b mod m)", (as in the Mathematica program below):
   If f(n)=(n mod 2), then (a,b,a,b,a,b,...) is given by
   a*f(n+1)+b*f(n), so that "(a or b mod m)" is given by
   a*f(n+1)+b*f(n)+m*floor((n-1)/2)), for n>=1.
This sequence is a permutation of the natural numbers.  - L. Edson Jeffery, Aug 13 2014

			Northwest corner:
1...3...7....15...31
2...4...8....16...32
5...11..23...47...95
6...12..24...48...96
9...19..39...79...159

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A042963, A014601, A191665, A191426.

Cf. A241957.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a = 3; b = 4; m[n_] := If[Mod[n, 2] == 0, 1, 0];
f[n_] := a*m[n + 1] + b*m[n] + 4*Floor[(n - 1)/2]
Table[f[n], {n, 1, 30}]  (* A014601(n+2): (4+4k,5+4k) *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191664 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191664  *)
(* Clark Kimberling, Jun 11 2011 *)
Grid[Table[2^k*(2*Floor[(n + 1)/2] - 1) - Mod[n, 2], {n, 12}, {k, 12}]] (* L. Edson Jeffery, Aug 13 2014 *)

A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.

Original entry on oeis.org

1, 3, 5, 7, 11, 9, 15, 23, 19, 13, 31, 47, 39, 27, 17, 63, 95, 79, 55, 35, 21, 127, 191, 159, 111, 71, 43, 25, 255, 383, 319, 223, 143, 87, 51, 29, 511, 767, 639, 447, 287, 175, 103, 59, 33, 1023, 1535, 1279, 895, 575, 351, 207, 119, 67, 37

Offset: 1

L. Edson Jeffery, Aug 09 2014

The sequence is a permutation of the odd natural numbers, since A(n,k) = 2*A054582(n-1,k-1) - 1 and A054582 is a permutation of the natural numbers.
For j a natural number, 2*j - 1 appears in row A001511(j) of A.
This is the square array A075300 with the first row omitted. - Peter Bala, Feb 07 2017

			Array begins:
.      1     5     9    13    17     21     25     29     33     37
.      3    11    19    27    35     43     51     59     67     75
.      7    23    39    55    71     87    103    119    135    151
.     15    47    79   111   143    175    207    239    271    303
.     31    95   159   223   287    351    415    479    543    607
.     63   191   319   447   575    703    831    959   1087   1215
.    127   383   639   895  1151   1407   1663   1919   2175   2431
.    255   767  1279  1791  2303   2815   3327   3839   4351   4863
.    511  1535  2559  3583  4607   5631   6655   7679   8703   9727
.   1023  3071  5119  7167  9215  11263  13311  15359  17407  19455

Vincenzo Librandi, Rows n = 0..50, flattened
Index entries for sequences related to 3x+1 (or Collatz) problem.

Cf. A016813, A017101 (rows 1 and 2).
Cf. A000225, A083329, A153894, A086224, A052996, etc. (columns 1-5).
Cf. A005408 (odd natural numbers), A054582.
Cf. A075300.

(* Array: *)
Grid[Table[2^n*(2*k - 1) - 1, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[2^(n - k + 1)*(2*k - 1) - 1, {n, 10}, {k, n}]]

A(n,k) = 2*A054582(n-1,k-1) - 1.

A249715 Positions of records in A249695.

Original entry on oeis.org

0, 4, 6, 13, 15, 27, 47, 55, 111, 223, 447, 895, 1791, 3583, 7167, 14335, 28671, 57343, 114687, 229375, 458751, 917503, 1835007, 3670015, 7340031, 14680063, 29360127, 58720255, 117440511

Offset: 1

Charles R Greathouse IV & Antti Karttunen, Nov 04 2014

nonn

From n=8 [a(8)=55] onward, the terms seem to be given by A086224(n-5), i.e. as (7 * 2^(n-5)) - 1.

A249714 gives the corresponding record values.

Cf. A249695, A086224, A249440.

A176448 a(n) = 7*2^n - 2.

Original entry on oeis.org

5, 12, 26, 54, 110, 222, 446, 894, 1790, 3582, 7166, 14334, 28670, 57342, 114686, 229374, 458750, 917502, 1835006, 3670014, 7340030, 14680062, 29360126, 58720254, 117440510, 234881022, 469762046, 939524094, 1879048190, 3758096382

Offset: 0

Vincenzo Librandi, Apr 18 2010

a(n) = 2*A086224(n-1) for n > 0 [Berselli - Librandi]. - Vincenzo Librandi, Aug 27 2010

			a(1) = 2*(5 + 1) = 12;
a(2) = 2*(12 + 1) = 26;
a(3) = 2*(26 + 1) = 54.

Index entries for linear recurrences with constant coefficients, signature (3,-2).

Magma
```
[7*2^n-2: n in [0..100]]
```

Table[7*2^n-2,{n,0,30}] (* or *) LinearRecurrence[{3,-2},{5,12},30] (* Harvey P. Dale, May 21 2017 *)

a(n) = 2*(a(n-1) + 1) with a(0)=5.
From R. J. Mathar, May 02 2010: (Start)
a(n)= 3*a(n-1) - 2*a(n-2).
G.f.: ( 5-3*x ) / ( (2*x-1)*(x-1) ). (End)
a(n) = A174317(n+3). - R. J. Mathar, Feb 21 2016

A267604 Decimal representation of the middle column of the "Rule 175" elementary cellular automaton starting with a single ON (black) cell.

Original entry on oeis.org

1, 3, 6, 13, 27, 55, 111, 223, 447, 895, 1791, 3583, 7167, 14335, 28671, 57343, 114687, 229375, 458751, 917503, 1835007, 3670015, 7340031, 14680063, 29360127, 58720255, 117440511, 234881023, 469762047, 939524095, 1879048191, 3758096383, 7516192767

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. A086224, A265186, A266678, A266680.

rule=175; 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 *) mc=Table[catri[[k]][[k]],{k,1,rows}]; (* Keep only middle cell from each row *) Table[FromDigits[Take[mc,k],2],{k,1,rows}]  (* Binary Representation of Middle Column *)

Conjectures from Colin Barker, Jan 18 2016 and Apr 20 2019: (Start)
a(n) = 3*a(n-1)-2*a(n-2) for n>3.
G.f.: (1-x^2+x^3) / ((1-x)*(1-2*x)). (End)
Conjectures from Altug Alkan, Jan 18 2016: (Start)
a(n) = A086224(n-2), for n > 1.
a(n) = ceiling((7/4)*2^n - 1). (End)

A277215 a(n) is the smallest even number not congruent to 1 modulo 3 that starts a (2n+1)-element alternating sequence of x/2 and (3x+1) iterations ending in the maximum of its Collatz trajectory.

Original entry on oeis.org

0, 26, 6, 14, 30, 1214, 1662, 254, 510, 1022, 2046, 28670, 40958, 180222, 32766, 65534, 131070, 1835006, 5767166, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 469762046, 671088638, 268435454, 536870910, 7516192766, 2147483646, 4294967294, 8589934590, 17179869182, 34359738366, 755914244094

Offset: 0

Hartmut F. W. Hoft, Nov 03 2016

nonn

a(n) starts a maximal alternating Collatz sequence v_0, ..., v_2n of 2n+1 elements and must have the form v_0 = 2*(q*2^n - 1) where q is the smallest odd number not a multiple of 3 such that v_(2n) = 2*(q*3^n - 1) is the maximum of its Collatz trajectory.
The intermediate elements of the sequence for 1 <= j <= n are v_(2j-1) = q * 2^(n-j+1) * 3^(j-1) - 1, which is odd, and v_(2j) = 2 * (q * 2^(n-j) * 3^j - 1), which is congruent to 2 modulo 4 except for j=n.
A277875(n) is the odd multiplier q in the expression for a(n).
Subsequences of a(n) are related to subsequences of the following sequences depending on the value of q:
a(n) = 2*A000225(n) = A000918(n+1) when A277875(n) = 1;
a(n) = 2*A153894(n) = A131051(n+3) when A277875(n) = 5;
a(n) = 2*A086224(n) = A086224(n+1)-1 = A176448(n+1) when A277875(n) = 7;
a(n) = A086225(n+1)-1 when A277875(n) = 11;
a(n) = A198274(n+1)-1 when A277875(n) = 13;
a(n) = A198276(n+1)-1 when A277875(n) = 19;
For small q > 1, the positions of 2*(q*2^n - 1) among the first 200 numbers in the sequence are:
q = 5: 12, 26, 36, 46, 58, 62, 174;
q = 7: 1, 11, 17, 25, 29, 45, 49, 53, 57, 61, 65, 77, 93, 103, 109, 113, 117, 125, 139, 141, 145, 157, 165, 173, 187, 189, 193;
q = 11: 13, 18, 35, 59, 69, 75, 83, 114, 133, 179;
q = 13: 6, 118;
q = 19: 5;
and among the first 400 numbers are:
q = 17: 222, 229, 230, 268;
(see A277875).
Conjecture: For every n there is an odd number q such that the alternating sequence ends in v_(2n), the maximum of the Collatz trajectory of a(n)=v_0.

			a(0) = 0 = 2*(1*2^0 - 1) since it is the start and end of the first alternating sequence of 1 element and the maximum of its trajectory.
a(1) = 26 = 2*(7*2^1 - 1) since sequence 26, 13, 40 has 3 elements and ends in the maximum of its trajectory and since 2, 10 and 18 do not satisfy the conditions for a(1).
a(5) = 1214 = 2*(19*2^5 - 1) starts the alternating sequence of 11 elements - 1214, 607, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232 - that ends in the trajectory maximum 9232 while the 11-element alternating sequences starting at 2*(q*2^5 - 1) with odd q<19 either do not end at the trajectory maximum or are congruent to 1 modulo 3 and therefore do not have maximal length.

Cf. A000225, A000918, A086224, A086225, A131051, A153894, A176448, A198274, A198276, A277875.

collatz[n_] := If[OddQ[n], 3n+1, n/2]
altdata[low_, high_] := Module[{n, q, notDone, v, a, m, list={}}, For[n=low, n<=high, n++, q=-1; notDone=True; While[notDone, q+=2; v=2*(q*2^n-1); If[Mod[v, 3]!=1, a=NestWhile[collatz, v, Mod[#,4]!=0&]; m=Max[NestWhileList[collatz, a, #!=1&]]; notDone=(a!=m)]]; AppendTo[list, {n, q, v, a}]]; list]/;(low>1)
a277215[n_]:=Map[#[[3]]&, altdata[2,n]]
Join[{0,26}, a277215[35]] (* sequence data *)

A332998 a(n) = A000120(A332996(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2, 3, 1, 1, 2, 2, 0, 1, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 1, 2, 1, 2, 2, 2, 3, 4, 1, 2, 1, 1, 2, 3, 2, 3, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2, 3, 1, 1, 2, 2, 0, 1, 0, 0, 1, 2, 1, 2, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 2, 3, 1, 1, 2, 2, 2, 3

Offset: 0

Antti Karttunen, Mar 05 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

Cf. A000120, A332817, A332898, A332996, A332997.

Cf. A086224.

A332998(n) = hammingweight(A332996(n));

PARI
```
A332998(n) = A332898(A332817(n));
```

a(n) = A000120(A332996(n)) = A332898(A332817(n)).
a(n) = A000120(n) - A332997(n).
Conjecture: for n >= 2, a(A086224(n-1)) = n, this being also the first occurrence of each n larger than 1.

A050523 Primes of the form 7*2^k - 1.

Original entry on oeis.org

13, 223, 3583, 917503, 14680063, 3758096383, 246290604621823, 1340933598257652751063553648756520535666396731910651903

Offset: 1

N. J. A. Sloane, Dec 29 1999

Index entries for sequences of n such that k*2^n-1 (or k*2^n+1) is prime.

See A001771 for more terms.

Cf. A086224, A158795 [Vincenzo Librandi, Mar 28 2009]

Select[Table[7*2^n-1, {n,0,200}], PrimeQ] (* Jean-François Alcover, Mar 20 2011 *)

a(n) = A086224(A001771(n)). - Elmo R. Oliveira, Apr 22 2025

13 inserted by R. J. Mathar, Apr 01 2009

cp's OEIS Frontend

A198276 a(n) = 19*2^n-1.

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A171389 a(n) = 21*2^n - 1.

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A191664 Dispersion of A014601 (numbers >2, congruent to 0 or 3 mod 4), by antidiagonals.

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A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n*(2*k - 1) - 1, n,k >= 1.

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A249715 Positions of records in A249695.

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A176448 a(n) = 7*2^n - 2.

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A267604 Decimal representation of the middle column of the "Rule 175" elementary cellular automaton starting with a single ON (black) cell.

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A277215 a(n) is the smallest even number not congruent to 1 modulo 3 that starts a (2n+1)-element alternating sequence of x/2 and (3x+1) iterations ending in the maximum of its Collatz trajectory.

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A241957 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = 2^n(2k - 1) - 1, n,k >= 1.