A130198 -id:A130198 - cp's OEIS frontend

A165211 Period 8: repeat [0,1,0,1,1,0,1,0].

0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0

Offset: 0

Philippe Deléham, Sep 07 2009

Parity of A064706.
Parity of the generalized pentagonal numbers A001318. - Omar E. Pol, Feb 04 2012
More generally, parity of the generalized k-gonal numbers, for odd k >= 5. - Omar E. Pol, Feb 05 2012

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).

Cf. A001318, A002817, A064706.

Cf. A130198 (essentially the same).

PadRight[{},112,{0,1,0,1,1,0,1,0}] (* Harvey P. Dale, Jan 29 2012 *)
Table[Mod[n*(n+1)*(n^2+n+2)/8,2],{n,0,100}] (* Vaclav Kotesovec, Apr 28 2014 after Wesley Ivan Hurt *)

a(n)=bitxor(n, n\4)%2 \\ Charles R Greathouse IV, Jul 13 2016

concat(0, Vec(x*(1 - x + x^2) / ((1 - x)*(1 + x^4)) + O(x^100))) \\ Colin Barker, Dec 20 2017

def A165211(n): return n&1^bool(n&4) # Chai Wah Wu, Aug 30 2024

a(n) = A002817(n) mod 2. - Wesley Ivan Hurt, Apr 23 2014
a(n) = 1/2 - (-1)^(n*(n+1)*(n^2 + n + 2)/8)/2. - Vaclav Kotesovec, Apr 28 2014
From Colin Barker, Dec 20 2017: (Start)
G.f.: x*(1 - x + x^2) / ((1 - x)*(1 + x^4)).
a(n) = a(n-1) - a(n-4) + a(n-5) for n>4.
(End)

A132194 a(n) = 1 if n-th prime is 0 or 2 mod 3, otherwise 0.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0

Offset: 1

Roger L. Bagula, Nov 05 2007

Equivalently, a(n) = 0 if n-th prime is 1 mod 3, otherwise 1. - Wouter Meeussen, May 21 2019

Binary sequence based on the primes: play it at a slower tempo to appreciate the irregularities.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for characteristic functions.

Characteristic function of A133677.

Cf. A039701, A130198, A099618.

[(NthPrime(n) mod 3) eq 1 select 0 else 1: n in [1..200]]; // G. C. Greubel, May 21 2019

a := n -> 1 - irem(modp(ithprime(n), 3), 2):
seq(a(n), n = 1..105); # Peter Luschny, May 21 2019

Table[If[Mod[Prime[n],3]== 1,0,1],{n,200}] (* Harvey P. Dale, May 21 2019 *)

{a(n) = if(prime(n)%3==1, 0, 1)}; \\ G. C. Greubel, May 21 2019

def a(n):
    if (mod(nth_prime(n), 3)==1): return 0
    else: return 1
[a(n) for n in (1..200)] # G. C. Greubel, May 21 2019

a(n) = 1-A099618(n). - R. J. Mathar, Jun 06 2019

Sum_{k=1..n} a(k) ~ n / 2. - Amiram Eldar, Mar 14 2025

Definition corrected by Harvey P. Dale, May 21 2019

A374646 Paradiddle version of Thue-Morse sequence.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1

Offset: 0

Robert P. P. McKone, Jul 15 2024

A paradiddle is a basic drum pattern, either "left left right left" or "right right left right". We can take left, right to be either 0, 1 or 1, 0.

Limiting word of the morphism with maps 0 |--> 0100, 1 |--> 1011 and axiom 1011. - Joerg Arndt, Jul 15 2024

			k = 0: Sequence starts at its simplest form;
1.
-----------------------------------------------
k = 1: The 1 of the initial sequence expands following the morphism rules, where 1 -> {1, 0, 1, 1} and 0 -> {0, 1, 0, 0}, resulting in;
1, 0, 1, 1.
-----------------------------------------------
k = 2: Each element of the initial sequence expands following the morphism rules, where 1 -> {1, 0, 1, 1} and 0 -> {0, 1, 0, 0};
1, 0, 1, 1,
0, 1, 0, 0,
1, 0, 1, 1,
1, 0, 1, 1.
-----------------------------------------------
k = 3: The expansion is applied recursively, giving:
1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1,
0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0,
1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1,
1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1.

Robert P. P. McKone, Table of n, a(n) for n = 0..16383
Index entries for sequences that are fixed points of mappings

Cf. A160381, A130198 (single paradiddle), A010059, A010060, A374724.

SubstitutionSystem[{1 -> {1, 0, 1, 1}, 0 -> {0, 1, 0, 0}}, {1}, {4}] // Flatten

first(n,v=[1])=if(n>4*#v, v=first((n+3)\4)); my(u=List()); for(i=1,#v-1, listput(u,v[i]); listput(u,1-v[i]); listput(u,v[i]); listput(u,v[i])); my(t=vector(n-#u,i,if(i==2,1-v[#v],v[#v]))); for(j=1,#t, listput(u,t[j])); Vec(u) \\ Charles R Greathouse IV, Jul 31 2024

a(n) = A160381(n)+1 mod 2. - Kevin Ryde, Dec 28 2024

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A165211 Period 8: repeat [0,1,0,1,1,0,1,0].

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A132194 a(n) = 1 if n-th prime is 0 or 2 mod 3, otherwise 0.

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A374646 Paradiddle version of Thue-Morse sequence.

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