A052940 -id:A052940 - cp's OEIS frontend

A083329 a(0) = 1; for n > 0, a(n) = 3*2^(n-1) - 1.

1, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943

Offset: 0

Paul Barry, Apr 27 2003

Apart from leading term (which should really be 3/2), same as A055010.
Binomial transform of A040001. Inverse binomial transform of A053156.
a(n) = A105728(n+1,2). - Reinhard Zumkeller, Apr 18 2005
Row sums of triangle A133567. - Gary W. Adamson, Sep 16 2007
Row sums of triangle A135226. - Gary W. Adamson, Nov 23 2007
a(n) = number of partitions Pi of [n+1] (in standard increasing form) such that the permutation Flatten[Pi] avoids the patterns 2-1-3 and 3-1-2. Example: a(3)=11 counts all 15 partitions of [4] except 13/24, 13/2/4 which contain a 2-1-3 and 14/23, 14/2/3 which contain a 3-1-2. Here "standard increasing form" means the entries are increasing in each block and the blocks are arranged in increasing order of their first entries. - David Callan, Jul 22 2008
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 42, 138, 162, 168, lead to this sequence. For the central square these vectors lead to the companion sequence A003945. - Johannes W. Meijer, Aug 15 2010
The binary representation of a(n) has n+1 digits, where all digits are 1's except digit n-1. For example: a(4) = 23 = 10111 (2). - Omar E. Pol, Dec 02 2012
Row sums of triangle A209561. - Reinhard Zumkeller, Dec 26 2012
If a Stern's sequence based enumeration system of positive irreducible fractions is considered (for example, A007305/A047679, A162909/A162910, A071766/A229742, A245325/A245326, ...), and if it is organized by blocks or levels (n) with 2^n terms (n >= 0), and the fractions, term by term, are summed at each level n, then the resulting sequence of integers is a(n) + 1/2, apart from leading term (which should be 1/2). - Yosu Yurramendi, May 23 2015
For n >= 2, A083329(n) in binary representation is a string [101..1], also 10 followed with (n-1) 1's. For n >= 3, A036563(n) in binary representation is a string [1..101], also (n-2) 1's followed with 01. Thus A083329(n) is a reflection of the binary representation of A036563(n+1). Example: A083329(5) = 101111 in binary, A036563(6) = 111101 in binary. - Ctibor O. Zizka, Nov 06 2018
For n > 0, a(n) is the minimum number of turns in (n+1)-dimensional Euclidean space needed to visit all 2^(n+1) vertices of the (n+1)-cube (e.g., {0,1}^(n+1)) and return to the starting point, moving along straight-line segments between turns (turns may occur elsewhere in R^(n+1)). - Marco Ripà, Aug 14 2025

			a(0) = (3*2^0 - 2 + 0^0)/2 = 2/2 = 1 (use 0^0=1).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5.
Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
Agustín Moreno Cañadas, Hernán Giraldo, Gabriel Bravo Rios, On the Number of Sections in the Auslander-Reiten Quiver of Algebras of Dynkin Type, Far East Journal of Mathematical Sciences (FJMS), Vol. 101, No. 8 (2017), pp. 1631-1654.
Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Essentially the same as A055010 and A052940.
Cf. A000225, A052955, A133567, A135226.
Cf. A007505 (primes).
Cf. A266550 (independence number of the n-Mycielski graph).

a083329 n = a083329_list !! n
a083329_list = 1 : iterate ((+ 1) . (* 2)) 2
-- Reinhard Zumkeller, Dec 26 2012, Feb 22 2012

[1] cat [3*2^(n-1)-1: n in [1..40]]; // Vincenzo Librandi, Jan 01 2016

seq(ceil((2^i+2^(i+1)-2)/2), i=0..31); # Zerinvary Lajos, Oct 02 2007

a[1] = 2; a[n_] := 2a[n - 1] + 1; Table[ a[n], {n, 31}] (* Robert G. Wilson v, May 04 2004 *)
Join[{1}, LinearRecurrence[{3, -2}, {2, 5}, 40]] (* Vincenzo Librandi, Jan 01 2016 *)

a(n)=(3*2^n-2+0^n)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = (3*2^n - 2 + 0^n)/2.
G.f.: (1-x+x^2)/((1-x)*(1-2*x)). [corrected by Martin Griffiths, Dec 01 2009]
E.g.f.: (3*exp(2*x) - 2*exp(x) + exp(0))/2.
a(0) = 1, a(n) = sum of all previous terms + n. - Amarnath Murthy, Jun 20 2004
a(n) = 3*a(n-1) - 2*a(n-2) for n > 2, a(0)=1, a(1)=2, a(2)=5. - Philippe Deléham, Nov 29 2013
From Bob Selcoe, Apr 25 2014: (Start)
a(n) = (...((((((1)+1)*2+1)*2+1)*2+1)*2+1)...), with n+1 1's, n >= 0.
a(n) = 2*a(n-1) + 1, n >= 2.
a(n) = 2^n + 2^(n-1) - 1, n >= 2. (End)
a(n) = A086893(n) + A061547(n+1), n > 0. - Yosu Yurramendi, Jan 16 2017

A055010 a(0) = 0; for n > 0, a(n) = 3*2^(n-1) - 1.

Original entry on oeis.org

0, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

Henry Bottomley, May 31 2000

Apart from leading term (which should really be 3/2), same as A083329.
Written in binary, a(n) is 1011111...1.
The sequence 2, 5, 11, 23, 47, 95, ... apparently gives values of n such that Nim-factorial(n) = 2. Cf. A059970. However, compare A060152. More work is needed! - John W. Layman, Mar 09 2001
With offset 1, number of (132,3412)-avoiding two-stack sortable permutations.
Number of descents after n+1 iterations of morphism A007413.
a(n) = A164874(n,1), n>0; subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-1). - Milan Janjic, Jan 24 2010
a(n) is the total number of records over all length n binary words. A record in a word a_1,a_2,...,a_n is a letter a_j that is larger than all the preceding letters. That is, a_j>a_i for all iGeoffrey Critzer, Jul 18 2020
Called Thabit numbers after the Syrian mathematician Thābit ibn Qurra (826 or 836 - 901). - Amiram Eldar, Jun 08 2021
a(n) is the number of objects in a pile that represents a losing position in a Nim game, where a player must select at least one object but not more than half of the remaining objects, on their turn. - Kiran Ananthpur Bacche, Feb 03 2025

			a(3) = 3*2^2 - 1 = 3*4 - 1 = 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric S. Egge and Toufik Mansour, 132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers, arXiv:math/0205206 [math.CO], 2002.
Sergey Kitaev and Toufik Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
Hunar Sherzad Taher and Saroj Kumar Dash, On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 448-459. See p. 2.
Eric Weisstein's World of Mathematics, Thabit ibn Kurrah Number.
Wikipedia, Thabit number.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A007505 for primes in this sequence. Apart from initial term, same as A052940 and A083329.
Cf. A266550 (independence number of the n-Mycielski graph).
Cf. A000079, A000225, A007283, A007413, A010036, A030130, A033484, A059970, A060152, A099258, A118654, A164874, A196168.

Concatenation([0], List([1..35], n-> 3*2^(n-1)-1)); # G. C. Greubel, May 06 2019

[Floor(3*2^(n-1) - 1): n in [0..35]]; // Vincenzo Librandi, May 18 2011

Join[{0},3*2^Range[0,34]-1] (* Harvey P. Dale, May 05 2013 *)

a(n)=3*2^n\2 - 1 \\ Charles R Greathouse IV, Apr 08 2016

[0]+[3*2^(n-1)-1 for n in (1..35)] # G. C. Greubel, May 06 2019

a(n) = A118654(n-1, 4), for n > 0.
a(n) = 2*a(n-1) + 1 = a(n-1) + A007283(n-1) = A007283(n)-1 = A000079(n) + A000225(n + 1) = A000079(n + 1) + A000225(n) = 3*A000079(n) - 1 = 3*A000225(n) + 2.
a(n) = A010036(n)/2^(n-1). - Philippe Deléham, Feb 20 2004
a(n) = A099258(A033484(n)-1) = floor(A033484(n)/2). - Reinhard Zumkeller, Oct 09 2004
G.f.: x*(2-x)/((1-x)*(1-2*x)). - Philippe Deléham, Oct 04 2011
a(n+1) = A196168(A000079(n)). - Reinhard Zumkeller, Oct 28 2011
E.g.f.: (3*exp(2*x) - 2*exp(x) - 1)/2. - Stefano Spezia, Sep 14 2024

A086224 a(n) = 7*2^n - 1.

Original entry on oeis.org

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, 15032385535, 30064771071

Offset: 0

Marco Matosic, Jul 27 2003

a(n) = A164874(n+2,2); subsequence of A030130. - Reinhard Zumkeller, Aug 29 2009

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=-3, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^n*charpoly(A,-5). - Milan Janjic, Jan 27 2010

Michael De Vlieger, Table of n, a(n) for n = 0..3319
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. 3-5, 14.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Other sequences with recurrence a(n+1) = 2*a(n) + 1 are:
a(0) = 2 gives A153893, a(0)=3 essentially A126646.
a(0) = 4 gives A153894, a(0)=5 essentially A153893.
a(0) = 7 gives essentially A000225.
a(0) = 8 gives A052996 except for some initial terms,
a(0) = 9 is essentially A153894.
a(0) = 10 gives A086225,
a(0) = 11 is essentially A153893.
a(0) = 13 is essentially A086224.
Cf. A030130, A052940, A164874.

7*2^Range[0,30]-1 (* Harvey P. Dale, May 09 2018 *)

a(n)=7<Charles R Greathouse IV, Sep 24 2015

a(n+1) = 2*a(n) + 1.
G.f.: (6-5*x)/((1-x)*(1-2*x)). - Jaume Oliver Lafont, Sep 14 2009
a(n-1)^2 + a(n) = (7*2^(n-1))^2. - Vincenzo Librandi, Aug 08 2010
a(n) = (A052940(n+1) + A000225(n+3))/2. - Gennady Eremin, Aug 31 2023
From Elmo R. Oliveira, Apr 22 2025: (Start)
E.g.f.: exp(x)*(7*exp(x) - 1).
a(n) = 3*a(n-1) - 2*a(n-2). (End)

More terms from David Wasserman, Feb 22 2005

A036991 Numbers k with the property that in the binary expansion of k, reading from right to left, the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

0, 1, 3, 5, 7, 11, 13, 15, 19, 21, 23, 27, 29, 31, 39, 43, 45, 47, 51, 53, 55, 59, 61, 63, 71, 75, 77, 79, 83, 85, 87, 91, 93, 95, 103, 107, 109, 111, 115, 117, 119, 123, 125, 127, 143, 151, 155, 157, 159, 167, 171, 173, 175, 179, 181, 183, 187, 189, 191, 199, 203

Offset: 1

N. J. A. Sloane

List of binary words that correspond to a valid pairing of parentheses. - Joerg Arndt, Nov 27 2004
This sequence includes as subsequences A000225, A002450, A007583, A036994, A052940, A112627, A113836, A113841, A290114; and also A015521 (without 0), A083713 (without 0), A086224 (without 6), A182512 (without 0). - Gennady Eremin, Nov 27 2021 and Aug 26 2023
Partial differences are powers of 2 (cf. A367626, A367627). - Gennady Eremin, Dec 23 2021
This is the sequence A030101(A014486(n)), n >= 0, sorted into ascending order. See A014486 for more references, illustrations, etc., concerning Dyck paths and other associated structures enumerated by the Catalan numbers. - Antti Karttunen, Sep 25 2023
The terms in this sequence with a given length in base 2 are counted by A001405. For example, the number of terms of bit length k=5 (these are 19, 21, 23, 27, 29, and 31) is equal to A001405(k-1) = A001405(4) = 6. - Gennady Eremin, Nov 07 2023

			From _Joerg Arndt_, Dec 05 2021: (Start)
List of binary words with parentheses for those in the sequence (indicated by P). The binary words are scanned starting from the least significant bit, while the parentheses words are written left to right:
     Binary   Parentheses (if the value is in the sequence)
00:  ..... P  [empty string]
01:  ....1 P   ()
02:  ...1.
03:  ...11 P   (())
04:  ..1..
05:  ..1.1 P   ()()
06:  ..11.
07:  ..111 P   ((()))
08:  .1...
09:  .1..1
10:  .1.1.
11:  .1.11 P   (()())
12:  .11..
13:  .11.1 P   ()(())
14:  .111.
15:  .1111 P   (((())))
16:  1....
17:  1...1
18:  1..1.
19:  1..11 P   (())()
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..13496 (all terms < 2^16; first 1000 terms from Reinhard Zumkeller)
Joerg Arndt, Matters Computational (The Fxtbook), section 1.28, pp. 78-80.
Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318, 2023. See pp. 1-2, 4, 7-11, 13-14.
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 2-5, 14.
Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
Index entries for sequences related to binary expansion of n

Cf. A350577 (primes subsequence).
See also A014486, A030101, A036988, A036990, A036992. A036994 is a subset (requires the count of zeros to be strictly less than the count of 1's).
See also A030308, A000225, A002450, A007583, A350346, A367625, A367626 & A367627 (first differences).

a036991 n = a036991_list !! (n-1)
a036991_list = filter ((p 1) . a030308_row) [0..] where
   p     []    = True
   p ones (0:bs) = ones > 1 && p (ones - 1) bs
   p ones (1:bs) = p (ones + 1) bs
-- Reinhard Zumkeller, Jul 31 2013

q:= proc(n) local l, t, i; l:= Bits[Split](n); t:=0;
      for i to nops(l) do t:= t-1+2*l[i];
        if t<0 then return false fi
      od: true
    end:
select(q, [$0..300])[];  # Alois P. Heinz, Oct 09 2019

moreOnesRLQ[n_Integer] := Module[{digits, len, flag = True, iter = 1, ones = 0, zeros = 0}, digits = Reverse[IntegerDigits[n, 2]]; len = Length[digits]; While[flag && iter < len, If[digits[[iter]] == 1, ones++, zeros++]; flag = ones >= zeros; iter++]; flag]; Select[Range[0, 203], moreOnesRLQ] (* Alonso del Arte, Sep 21 2011 *)
Join[{0},Select[Range[210],Min[Accumulate[Reverse[IntegerDigits[#,2]]/.{0->-1}]]>-1&]] (* Harvey P. Dale, Apr 18 2014 *)

select( {is_A036991(n,c=1)=!n||!until(!n>>=1,(c-=(-1)^bittest(n,0))||return)}, [0..99]) \\ M. F. Hasler, Nov 26 2021

def ok(n):
    if n == 0: return True # by definition
    count = {"0": 0, "1": 0}
    for bit in bin(n)[:1:-1]:
        count[bit] += 1
        if count["0"] > count["1"]: return False
    return True
print([k for k in range(204) if ok(k)]) # Michael S. Branicky, Nov 25 2021

from itertools import count, islice
def A036991_gen(): # generator of terms
    yield 0
    for n in count(1):
        s = bin(n)[2:]
        c, l = 0, len(s)
        for i in range(l):
            c += int(s[l-i-1])
            if 2*c <= i:
                break
        else:
            yield n
A036991_list = list(islice(A036991_gen(),20)) # Chai Wah Wu, Dec 30 2021

If a(n) = A000225(k) for some k, then a(n+1) = a(n) + A060546(k). - Gennady Eremin, Nov 07 2023

More terms from Erich Friedman
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar
Offset corrected and example adjusted accordingly by Reinhard Zumkeller, Jul 31 2013

A107909 Numbers having no consecutive zeros or no consecutive ones in binary representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 40, 41, 42, 43, 45, 46, 47, 53, 54, 55, 58, 59, 61, 62, 63, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 107, 109, 110, 111

Offset: 0

Reinhard Zumkeller, May 28 2005

Union of A003754 and A003714, complement of A107911;
a(A023548(n+2)) = A052940(n+1) for n>0;
a(A001924(n)) = A000225(n) = 2^n - 1;
a(A000126(n)) = A000079(n) = 2^n for n>0;
A107910(n) = a(n+1) - a(n).

Ivan Neretin, Table of n, a(n) for n = 0..10000
J. M. Dusel, Balanced parabolic quotients and branching rules for Demazure crystals, 2014.
Index entries for 2-automatic sequences.

Cf. A007088, A107907.

foreach $n(1..100){$_=sprintf("%b",$n); print "$n\n" if !m/11/||!m/00/}
# Ivan Neretin, May 01 2016

A208637 T(n,k)=Number of nXk 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

Original entry on oeis.org

1, 2, 2, 4, 5, 4, 8, 11, 13, 8, 16, 23, 32, 34, 16, 32, 47, 71, 95, 89, 32, 64, 95, 150, 225, 284, 233, 64, 128, 191, 309, 494, 722, 851, 610, 128, 256, 383, 628, 1042, 1652, 2331, 2552, 1597, 256, 512, 767, 1267, 2149, 3577, 5572, 7548, 7655, 4181, 512, 1024, 1535

Offset: 1

R. H. Hardin Feb 29 2012

Table starts
...1....2....4.....8....16.....32.....64....128.....256.....512....1024
...2....5...11....23....47.....95....191....383.....767....1535....3071
...4...13...32....71...150....309....628...1267....2546....5105...10224
...8...34...95...225...494...1042...2149...4375....8840...17784...35687
..16...89..284...722..1652...3577...7504..15448...31440...63543..127884
..32..233..851..2331..5572..12404..26508..55260..113427..230559..465773
..64..610.2552..7548.18888..43284..94320.199299..412962..844943.1714680
.128.1597.7655.24476.64216.151656.337227.722733.1512764.3117620.6359210

			Some solutions for n=4 k=3
..0..0..0....0..0..1....0..1..1....0..0..1....0..0..1....0..1..0....0..1..0
..1..1..0....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1....1..0..1
..0..1..1....0..1..0....0..1..0....0..1..0....1..0..0....0..1..0....1..0..1
..0..0..1....1..0..1....0..1..0....0..1..0....1..1..0....1..0..1....1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..1193

Column 2 is A001519(n+1)
Column 3 is A199109(n-1)
Row 2 is A052940(n-1)

Empirical for column k:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-2)
k=3: a(n) = 4*a(n-1) -3*a(n-2)
k=4: a(n) = 5*a(n-1) -6*a(n-2) +a(n-3)
k=5: a(n) = 6*a(n-1) -10*a(n-2) +4*a(n-3)
k=6: a(n) = 7*a(n-1) -15*a(n-2) +10*a(n-3) -a(n-4)
k=7: a(n) = 8*a(n-1) -21*a(n-2) +20*a(n-3) -5*a(n-4)
Empirical for row n:
n=1: a(k)=2*a(k-1)
n=2: a(k)=3*a(k-1)-2*a(k-2)
n=3: a(k)=4*a(k-1)-5*a(k-2)+2*a(k-3)
n=4: a(k)=5*a(k-1)-9*a(k-2)+7*a(k-3)-2*a(k-4) for k>5
n=5: a(k)=6*a(k-1)-14*a(k-2)+16*a(k-3)-9*a(k-4)+2*a(k-5) for k>7
n=6: a(k)=7*a(k-1)-20*a(k-2)+30*a(k-3)-25*a(k-4)+11*a(k-5)-2*a(k-6) for k>9
n=7: a(k)=8*a(k-1)-27*a(k-2)+50*a(k-3)-55*a(k-4)+36*a(k-5)-13*a(k-6)+2*a(k-7) for k>11

A336839 Denominator of the arithmetic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 07 2020

Also denominator of A336841(n) / A000005(n).

All terms are odd because A336932(n) = A007814(A003973(n)) >= A295664(n) for all n.

Cf. A000005, A000225, A003961, A003973, A007814, A052940, A057021, A295664, A336840, A336841, A336856, A336931, A336932.
Cf. A336918 (positions of 1's), A336919 (of terms > 1).
Cf. A336837 and A336838 (numerators).

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336839(n) = denominator(sigma(A003961(n))/numdiv(n));

a(n) = denominator(A003973(n)/A000005(n)).
a(n) = d(n)/A336856(n) = d(n)/gcd(d(n),A003973(n)) = d(n)/gcd(d(n),A336841(n)), where d(n) is the number of divisors of n, A000005(n).
a(n) = A057021(A003961(n)).
For all primes p, and e >= 0, a(A000225(e)) = a(p^((2^e) - 1)) = 1. [See A336856]
It seems that for all odd primes p, and with the exponents e=5, 11, 17 or 23 (at least these), a(p^e) = 1.
It seems that a(27^((2^n)-1)) = A052940(n-1) for all n >= 1.

A290114 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 3, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 0

Robert Price, Jul 19 2017

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..126
Robert Price, Diagrams of first 20 stages
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for linear recurrences with constant coefficients, signature (3,-2).
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Essentially the same as A153893, A083329, A055010, A052940, A266550.

Cf. A290111, A290112, A290113, .

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 643; stages = 128;
rule = IntegerDigits[code, 2, 10];
g = 2 * stages + 1; (* Maximum size of grid *)
a = PadLeft[{{1}}, {g, g}, 0,Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca = a;
ca = Table[ca = CAStep[rule, ca], {n, 1, stages + 1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k = (Length[ca[[1]]] + 1)/2;
ca = Table[Table[Part[ca[[n]] [[j]],Range[k + 1 - n, k - 1 + n]], {j, k + 1 - n, k - 1 + n}], {n, 1, k}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

For n>1, a(n) = 3*2^(n-1)-1.
a(n) = A266550(n+2) for n > 1. - Georg Fischer, Oct 30 2018
a(n) = 2*a(n-1) + 1 for n=1 and n>=3. - Gennady Eremin, Aug 26 2023
From Chai Wah Wu, Apr 02 2024: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) for n > 3.
G.f.: (2*x^3 - 2*x^2 + 1)/((x - 1)*(2*x - 1)). (End)

A134060 Triangle T(n,k) = A124927(n,k) + A134058(n,k) - A007318(n,k), read by rows.

Original entry on oeis.org

1, 2, 3, 2, 6, 3, 2, 9, 9, 3, 2, 12, 18, 12, 3, 2, 15, 30, 30, 15, 3, 2, 18, 45, 60, 45, 18, 3, 2, 21, 63, 105, 105, 63, 21, 3, 2, 24, 84, 168, 210, 168, 84, 24, 3, 2, 27, 108, 252, 378, 378, 252, 108, 27, 3

Offset: 0

Gary W. Adamson, Oct 05 2007

			First few rows of the triangle are:
  1;
  2,  3;
  2,  6,  3;
  2,  9,  9,  3;
  2, 12, 18, 12,  3;
  2, 15, 30, 30, 15, 3;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A007318, A052940 (row sums), A127927, A134058.

[1] cat [k eq 0 select 2 else 3*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021

Table[3*Binomial[n, k] -Boole[k==0] -Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)

def A134060(n,k): return 3*binomial(n,k) -bool(k==0) -bool(n==0)
flatten([[A134060(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021

T(n,k) = A124927(n,k) + A134058(n,k) - A007318(n,k) as infinite lower triangular matrices.
Sum_{k=0..n} T(n, k) = A052940(n).
T(n, k) = 3*binomial(n,k) - [k=0] - [n=0]. - G. C. Greubel, May 03 2021

A266550 Independence number of the n-Mycielski graph.

Original entry on oeis.org

1, 1, 2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471, 6442450943, 12884901887

Offset: 1

Eric W. Weisstein, Dec 31 2015

Eric Weisstein's World of Mathematics, Independence Number.
Eric Weisstein's World of Mathematics, Mycielski Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A052940, A055010, A083329, A153893.

[1,1] cat [-1+3*2^(n-3): n in [3..40]]; // Vincenzo Librandi, Jan 01 2016

I:=[1,1,2,5]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Jan 01 2016

Table[Piecewise[{{-1 + 3 2^(n - 3), n > 2}}, 1], {n, 35}]
CoefficientList[Series[1 + x*(1 - x + x^2)/((1 - x)*(1 - 2*x)), {x, 0, 35}], x] (* Vincenzo Librandi, Jan 01 2016 *)

a(1) = 1, a(2) = 1; for n>2, a(n) = -1 + 3*2^(n-3) = A083329(n-2) = A055010(n-2) = A153893(n-3).
G.f.: x + x^2*(1 - x + x^2)/((1 - x)*(1 - 2*x)).
a(n) = 3*a(n-1)-2*a(n-2) for n>2. - Vincenzo Librandi, Jan 01 2016
a(n) = A052940(n-3) for n > 3. - Georg Fischer, Oct 23 2018
E.g.f.: (3*exp(2*x) - 8*exp(x) + 5 + 10*x+ 2*x^2)/8. - Stefano Spezia, Sep 14 2024

cp's OEIS Frontend

A083329 a(0) = 1; for n > 0, a(n) = 3*2^(n-1) - 1.

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A055010 a(0) = 0; for n > 0, a(n) = 3*2^(n-1) - 1.

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A086224 a(n) = 7*2^n - 1.

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A036991 Numbers k with the property that in the binary expansion of k, reading from right to left, the number of 0's never exceeds the number of 1's.

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A107909 Numbers having no consecutive zeros or no consecutive ones in binary representation.

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A208637 T(n,k)=Number of nXk 0..1 arrays with new values 0..1 introduced in row major order and no element equal to more than one of its immediate leftward or upward or right-upward antidiagonal neighbors.

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A336839 Denominator of the arithmetic mean of the divisors of A003961(n).

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