A010674 -id:A010674 - cp's OEIS frontend

A021037 Duplicate of A010674.

0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0, 3, 0

Offset: 0

dead

A047208 Numbers that are congruent to {0, 4} mod 5.

Original entry on oeis.org

0, 4, 5, 9, 10, 14, 15, 19, 20, 24, 25, 29, 30, 34, 35, 39, 40, 44, 45, 49, 50, 54, 55, 59, 60, 64, 65, 69, 70, 74, 75, 79, 80, 84, 85, 89, 90, 94, 95, 99, 100, 104, 105, 109, 110, 114, 115, 119, 120, 124, 125, 129, 130, 134, 135, 139, 140, 144, 145, 149

Offset: 1

N. J. A. Sloane

Also solutions to 3^x + 5^x == 2 (mod 11). - Cino Hilliard, May 18 2003

G. C. Greubel, Table of n, a(n) for n = 1..5000
Cino Hilliard, solutions to 3^x + 5^x == 2 mod 11/, June 2003.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001622, A010674, A010685 (first differences), A274406.

[(5*(n-1) + 3*((n-1) mod 2))/2: n in [1..100]]; // G. C. Greubel, Nov 23 2021

{#,#+4}&/@(5*Range[0,30])//Flatten (* Harvey P. Dale, Apr 05 2019 *)

forstep(n=0,200,[4,1],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

[(5*(n-1) +3*((n-1)%2))/2 for n in (1..100)] # G. C. Greubel, Nov 23 2021

From R. J. Mathar, Jan 24 2009: (Start)
G.f.: x^2*(4+x)/((1-x)^2*(1+x)).
a(n) = a(n-2) + 5. (End)
a(n) = 5*n - 6 - a(n-1) (with a(1)=0). - Vincenzo Librandi, Nov 18 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*b(k), with b(0)=4 and b(k) = A020714(k-1) = 5*2^(k-1) for k>0. - Philippe Deléham, Oct 17 2011
a(n) = ceiling((5/3)*ceiling(3*n/2)). - Clark Kimberling, Jul 04 2012
a(n) = (5*(n-1) + 3*(n-1 mod 2))/2 = (5*(n-1) + A010674(n-1))/2. - G. C. Greubel, Nov 23 2021
Sum_{n>=2} (-1)^n/a(n) = log(5)/4 + log(phi)/(2*sqrt(5)) - sqrt(1+2/sqrt(5))*Pi/10, where phi is the golden ratio (A001622). - Amiram Eldar, Dec 07 2021
E.g.f.: 1 + ((5*x - 7/2)*exp(x) + (3/2)*exp(-x))/2. - David Lovler, Aug 23 2022

A350814 Numbers m such that the largest digit in the decimal expansion of 1/m is 3.

Original entry on oeis.org

3, 30, 33, 75, 300, 303, 330, 333, 429, 750, 813, 3000, 3003, 3030, 3125, 3300, 3330, 3333, 4290, 4329, 7500, 7575, 8130, 30000, 30003, 30030, 30300, 30303, 31250, 33000, 33300, 33330, 33333, 42900, 43290, 46875, 75000, 75075, 75750, 76923, 81103, 81300, 300000

Offset: 1

Bernard Schott, Jan 30 2022

If m is a term, 10*m is also a term.
3 is the only prime up to 2.6*10^8 (see comments in A333237).
Some subsequences:
{3, 30, 300, ...} = A093138 \ {1}.
{3, 33, 333, ...} = A002277 \ {0}.
{3, 33, 303, 3003, ...} = 3 * A000533.
{3, 303, 30303, 3030303, ...} = 3 * A094028.

			As 1/33 = 0.0303030303..., 33 is a term.
As 1/75 = 0.0133333333..., 75 is a term.
As 1/429 = 0.002331002331002331..., 429 is a term.

Index entries for sequences related to decimal expansion of 1/n

Cf. A000533, A094028, A266385, A333236.
Similar with largest digit k: A333402 (k=1), A341383 (k=2), A333237 (k=9).
Subsequences: A002277 \ {0}, A093138 \ {1}.
Decimal expansion: A010701 (1/3), A010674 (1/33).

Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 3 &] (* Amiram Eldar, Jan 30 2022 *)

from fractions import Fraction
from itertools import count, islice
from sympy import n_order, multiplicity
def repeating_decimals_expr(f, digits_only=False):
    """ returns repeating decimals of Fraction f as the string aaa.bbb[ccc].
        returns only digits if digits_only=True.
    """
    a, b = f.as_integer_ratio()
    m2, m5 = multiplicity(2,b), multiplicity(5,b)
    r = max(m2,m5)
    k, m = 10**r, 10**n_order(10,b//2**m2//5**m5)-1
    c = k*a//b
    s = str(c).zfill(r)
    if digits_only:
        return s+str(m*k*a//b-c*m)
    else:
        w = len(s)-r
        return s[:w]+'.'+s[w:]+'['+str(m*k*a//b-c*m)+']'
def A350814_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda m:max(repeating_decimals_expr(Fraction(1,m),digits_only=True)) == '3',count(max(startvalue,1)))
A350814_list = list(islice(A350814_gen(),10)) # Chai Wah Wu, Feb 07 2022

More terms from Amiram Eldar, Jan 30 2022

A010695 Period 2: repeat (2,5).

Original entry on oeis.org

2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5, 2, 5

Offset: 0

N. J. A. Sloane

Also decimal expansion of 25/99.
Continued fraction expansion of A176052. - R. J. Mathar, Mar 08 2012
Periodic part of the partial quotients of the continued fraction expansion of sqrt(7/5), which starts [1, 5, 2, 5, 2, 5, ...]. - Hugo Pfoertner, Jan 10 2025

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010674 (2^(1-(-1)^n) - 1).

Cf. A010691.

&cat [[2,5]^^50]; // Bruno Berselli, Dec 29 2015

PadRight[{}, 100, {2, 5}] (* Paolo Xausa, Jan 16 2025 *)

makelist(if evenp(n) then 2 else 5, n, 0, 80); /* Martin Ettl, Nov 09 2012 */

G.f.: (2+5*x)/((1-x)*(1+x)). - R. J. Mathar, Nov 21 2011
a(n) = 2^(1-(-1)^n) + 1. - Bruno Berselli, Dec 29 2015
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 10/a(n-1). See also A010691.
a(n) = 7 - a(n-1). See also A010702. (End)

Edited by Bruno Berselli, Dec 29 2015

A010698 Period 2: repeat (2,8).

Original entry on oeis.org

2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2

Offset: 0

N. J. A. Sloane

This is the regular (simple) continued fraction for (2+sqrt(5))/2 = A176055. - Antonia Redondo Buitrago, Jul 30 2009

Index entries for linear recurrences with constant coefficients, signature (0,1).

A010698:=n->5-3*(-1)^n; seq(A010698(n), n=0..100); # Wesley Ivan Hurt, Mar 26 2014

Table[5-3(-1)^n, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 26 2014 *)
PadRight[{},120,{2,8}] (* Harvey P. Dale, Oct 31 2016 *)

A010698(n):=if evenp(n) then 2 else 8$
makelist(A010698(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

a(n)=n%2*6+2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = -3*(-1)^n+5. - Paolo P. Lava, Oct 20 2006
G.f.: 2(1+4x)/((1-x)(1+x)). a(n) = 2*A010685(n). - R. J. Mathar, Oct 20 2008
a(n) = (A010674(n)+1)*2. - Martin Ettl, Nov 09 2012

A140253 a(2n) = 2(24^(n-1)-1) and a(2n-1) = 2*4^(n-1)-1.

Original entry on oeis.org

-1, 1, 2, 7, 14, 31, 62, 127, 254, 511, 1022, 2047, 4094, 8191, 16382, 32767, 65534, 131071, 262142, 524287, 1048574, 2097151, 4194302, 8388607, 16777214, 33554431, 67108862, 134217727, 268435454, 536870911

Offset: 0

Paul Curtz, Jun 23 2008

The inverse binomial transform is 1, 1, 4, -2, 10, -14, 34, -62 which leads to (-1)^(n+1)*A135440(n).
For n > 0: A266161(a(n)) = n and A266161(m) < n for m < a(n). - Reinhard Zumkeller, Dec 22 2015
Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 23 2017

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
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 sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2, 1, -2).

Cf. A000034, A000069, A000079, A002450, A010674, A014551, A083420, A266161.

import Data.List (transpose)
a140253 n = a140253_list !! n
a140253_list = -1 : concat
                    (transpose [a083420_list, map (* 2) a083420_list])
-- Reinhard Zumkeller, Dec 22 2015

A140253:=proc(n): if type(n, odd) then 2*4^(((n+1)/2)-1)-1 else 2*(2*4^((n/2)-1)-1) fi: end: seq(A140253(n),n=0..29); # Johannes W. Meijer, Jun 24 2011

Table[(2^(n+1) - 3 + (-1)^(n+1))/2, {n, 0, 30}] (* Jean-François Alcover, Jun 05 2017 *)

a(2*n) = 2*A083420(n-1) and a(2*n+1) = A083420(n)
a(n+1) - a(n) = A014551(n); Jacobsthal-Lucas numbers.
a(2*n) + a(2*n+1) = 9*A002450(n)
a(n+1) - 2*a(n) = A010674(n+1); repeat 3, 0.
a(n) + A000034(n+1) = A000079(n); powers of 2.
a(n)= a(n-1) + 2*a(n-2) + 3. - Gary Detlefs, Jun 22 2010
a(n+1) = A000069(2^n); odious numbers. - Johannes W. Meijer, Jun 24 2011
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2, a(0) = -1, a(1) = 1, a(2) = 2. - Philippe Deléham, Feb 25 2012
G.f.: (x^2+3*x-1)/((1-2*x)*(1-x)*(1+x)). - Philippe Deléham, Feb 25 2012

Edited, corrected and information added by Johannes W. Meijer, Jun 24 2011

A010680 Decimal expansion of 1/11.

Original entry on oeis.org

0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9, 0, 9

Offset: 0

N. J. A. Sloane

Period 2: repeat [0,9].

			1/11 = 0.0909090909090909090909090909090909090909090909090909090909...

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A000035, A010674.

Bisections give: A000004, A010734.

RealDigits[1/11, 10, 100][[1]] (* Alonso del Arte, Mar 11 2018 *)

a(n) = 9*(n%2); \\ Altug Alkan, Mar 25 2018

a(n) = (9/2)*(1 - (-1)^n) = 9*(n mod 2). - Paolo P. Lava, Oct 31 2006
From Elmo R. Oliveira, Jan 15 2024: (Start)
a(n) = a(n-2) for n >= 2.
a(n) = 3 * A010674(n).
G.f.: 9*x/(1-x^2).
E.g.f.: 9*sinh(x). (End)
a(n) = 9 * A000035(n). - Alois P. Heinz, Jan 16 2024

A168233 a(n) = 3*n - a(n-1) - 1 for n>0, a(1)=1.

Original entry on oeis.org

1, 4, 4, 7, 7, 10, 10, 13, 13, 16, 16, 19, 19, 22, 22, 25, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 46, 49, 49, 52, 52, 55, 55, 58, 58, 61, 61, 64, 64, 67, 67, 70, 70, 73, 73, 76, 76, 79, 79, 82, 82, 85, 85, 88, 88, 91, 91, 94, 94, 97, 97, 100, 100, 103, 103, 106

Offset: 1

Vincenzo Librandi, Nov 21 2009

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

Cf. A008585, A001477, A016777, A010674, A016789.

[(6*n + 3*(-1)^n + 1)/4: n in [1..70]]; // Vincenzo Librandi, Feb 02 2013

a:=n->3*floor(n/2)+1; seq(a(k), k = 1..70); # Wesley Ivan Hurt, Feb 01 2013

CoefficientList[Series[(1 + 3*x - x^2)/((1+x) * (1-x)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Feb 02 2013 *)
LinearRecurrence[{1,1,-1},{1,4,4},80] (* Harvey P. Dale, Oct 13 2015 *)

From Bruno Berselli, Nov 15 2010: (Start)
a(n) = (6*n + 3*(-1)^n + 1)/4.
G.f.: x*(1 + 3*x - x^2)/((1+x)*(1-x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3), for n>3.
a(n) + a(n-1) = A016789(n-1) for n>1.
a(n) - a(n-1-2*k) = A010674(n-1) + A008585(k) for n>2*k+1 and k in A001477.
a(n) - a(n-2*k) = A008585(k) for n>2*k and k in A001477. (End)
a(n+1) = A016777(floor((n+1)/2)). - R. J. Mathar, Jan 03 2011
E.g.f.: (1/4)*(3 - 4*exp(x) + (1 + 6*x)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 16 2016

A274912 Square array read by antidiagonals upwards in which each new term is the least nonnegative integer distinct from its neighbors.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Jul 11 2016

In the square array we have that:
Antidiagonal sums give A168237.
Odd-indexed rows give A010673.
Even-indexed rows give A010684.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed antidiagonals give the initial terms of A010674.
Even-indexed antidiagonals give the initial terms of A000034.
Main diagonal gives A010674.
This is also a triangle read by rows in which each new term is the least nonnegative integer distinct from its neighbors.
In the triangle we have that:
Row sums give A168237.
Odd-indexed columns give A000035.
Even-indexed columns give A010693.
Odd-indexed diagonals give A010673.
Even-indexed diagonals give A010684.
Odd-indexed rows give the initial terms of A010674.
Even-indexed rows give the initial terms of A000034.
Odd-indexed antidiagonals give the initial terms of A010673.
Even-indexed antidiagonals give the initial terms of A010684.

			The corner of the square array begins:
0, 2, 0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, 3, 1, ...
0, 2, 0, 2, 0, 2, ...
1, 3, 1, 3, 1, ...
0, 2, 0, 2, ...
1, 3, 1, ...
0, 2, ...
1, ...
...
The sequence written as a triangle begins:
0;
1, 2;
0, 3, 0;
1, 2, 1, 2;
0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2;
0, 3, 0, 3, 0, 3, 0, 3, 0;
1, 2, 1, 2, 1, 2, 1, 2, 1, 2;
...

Cf. A000034, A000035, A001477, A010673, A010674, A010684, A010693, A168237, A274913, A274920.

ListTools:-Flatten([seq([[0,3]$i,0,[1,2]$(i+1)],i=0..10)]); # Robert Israel, Nov 14 2016

Table[Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

a(n) = A274913(n) - 1.
From Robert Israel, Nov 14 2016: (Start)
G.f.: 3*x/(1-x^2) - Sum_{k>=0} (2*x^(2*k^2+3*k+1)-x^(2*k^2+5*k+3))/(1+x).
G.f. as triangle: x*(1+2*y+3*x*y)/((1-x^2*y^2)*(1-x^2)). (End)

A177499 Period 4: repeat [1, 16, 4, 16].

Original entry on oeis.org

1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16, 1, 16, 4, 16

Offset: 0

Paul Curtz, May 10 2010

From Klaus Brockhaus, May 14 2010: (Start)
Interleaving of A000012, A010855, A010709 and A010855.
Continued fraction expansion of (44+sqrt(2442))/88. (End)

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010674, A177838. [Klaus Brockhaus, May 14 2010]

Cf. A000012, A010709, A010855, A176895.

&cat[[1, 16, 4, 16]^^26]; // Klaus Brockhaus, May 14 2010

seq(op([1, 16, 4, 16]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

Table[{1, 16, 4, 16}, {21}] // Flatten (* Jean-François Alcover, May 24 2013 *)

From Klaus Brockhaus, May 14 2010: (Start)
a(n+2) - a(n) = A010674(n).
a(n) = a(n-4) for n > 3.
G.f.: (1+16*x+4*x^2+16*x^3)/(1-x^4). (End)
a(n) = A176895(n)^2. - Paul Curtz, Mar 21 2011
a(n) = (37 - 6*cos(n*Pi/2) - 27*cos(n*Pi) - 27*I*sin(n*Pi))/4. - Wesley Ivan Hurt, Jul 09 2016

cp's OEIS Frontend

A021037 Duplicate of A010674.

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A047208 Numbers that are congruent to {0, 4} mod 5.

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A350814 Numbers m such that the largest digit in the decimal expansion of 1/m is 3.

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A010695 Period 2: repeat (2,5).

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A010698 Period 2: repeat (2,8).

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A140253 a(2*n) = 2*(2*4^(n-1)-1) and a(2*n-1) = 2*4^(n-1)-1.

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A010680 Decimal expansion of 1/11.

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A140253 a(2n) = 2(24^(n-1)-1) and a(2n-1) = 2*4^(n-1)-1.