A083035 -id:A083035 - cp's OEIS frontend

A272532 Single bit representation of the sum of two sinusoids with periods 2 and 2*sqrt(2).

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

Offset: 1

Andres Cicuttin, May 02 2016

Since the ratio of the two periods is irrational, the sequence is strictly non-periodic.

From the factorized expression of the corresponding real function of x : 2*cos(2Pi((2 - sqrt(2))/8)x)*sin(2Pi((2 + sqrt(2))/8)x), it is possible to see that the largest distance between consecutive zeros is not greater than the shortest semi-period, 4/(2 + sqrt(2)), that is smaller than 2, and from this it follows that there are no more than two consecutive 0's or 1's.

Andres Cicuttin, Several segments at different scales of the fractal walk starting at (0,0) with step of unit length turning right if a(n)=1 and left if a(n)=0, except when this rule determines a decrement in the horizontal axes which in that case the step is equal to previous one

Conjectured quasiperiodicity in A271591 and A272170. A083035.

nmax=120 ; Table[If[Sin[2*Pi*(1/2)*n]+Sin[2*Pi*(1/(2*Sqrt[2]))*n]<0,0,1],{n,1,nmax}]

a(n) = floor( (1 + sin(2*Pi*(1/2)*n) + sin(2*Pi*(1/(2*Sqrt[2]))*n)) mod 2).

A176409 Multiples of 3 or 7.

Original entry on oeis.org

0, 3, 6, 7, 9, 12, 14, 15, 18, 21, 24, 27, 28, 30, 33, 35, 36, 39, 42, 45, 48, 49, 51, 54, 56, 57, 60, 63, 66, 69, 70, 72, 75, 77, 78, 81, 84, 87, 90, 91, 93, 96, 98, 99, 102, 105, 108, 111, 112, 114, 117, 119, 120, 123, 126, 129, 132, 133, 135, 138, 140, 141, 144, 147, 150, 153

Offset: 1

Zak Seidov, Dec 07 2010

Therefore, this sequence also includes multiples of 21.
First differences: 3, 3, 1, 2, 3, 2, 1, 3, 3, 3, 3, 1, 2, 3, 2, 1, 3, 3, 3, 3, 1, 2, 3, 2, 1, 3, 3, 3, 3, 1, 2, 3, 2, 1, 3, 3, 3 have period {3, 3, 1, 2, 3, 2, 1, 3, 3}.
In general, sequences of numbers divisible by primes p and q will be of the form a(n+p+q-1) = a(n)+p*q. - Gary Detlefs, Oct 07 2013

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A047229, A065520, A083035.

Union@Flatten@{Table[3n, {n, 70}], Table[7n, {n, 30}]}
Select[ Range@ 153, Mod[#, 3] == 0 || Mod[#, 7] == 0 &]

concat(0, Vec(x^2*(3 + 3*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6 + 3*x^7 + 3*x^8) / ((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)) + O(x^70))) \\ Colin Barker, Mar 21 2020

a(n+9) = a(n) + 21.
a(n) = 21*floor((n-1)/9) + 2*((n-1) mod 9) + s(((n-1) mod 9)-1) + 1 - floor(((n-2) mod 9)/8), where s(n) = floor(n*sqrt(2)) - 2*floor(n/sqrt(2)). - Gary Detlefs, Oct 07 2013
a(n) = 7n/3 + O(1). - Charles R Greathouse IV, Feb 13 2017
From Colin Barker, Mar 21 2020: (Start)
G.f.: x^2*(3 + 3*x + x^2 + 2*x^3 + 3*x^4 + 2*x^5 + x^6 + 3*x^7 + 3*x^8) / ((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)).
a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.
(End)

0 inserted by Daniel Starodubtsev, Mar 21 2020

A335617 Let c(1) = c(2) = 0, c(3) = 1, and c(n + 3) = (c(n) - 2*c(n + 1) + c(n + 2))/n, then a(n) = ceiling (c(n)).

Original entry on oeis.org

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

Offset: 0

Andres Cicuttin, Oct 11 2020

nonn

Conjectured quasiperiodicity with autocorrelation function R(x) = 1/2 if x = 0, 1/4 if x > 0.

Some other proved or conjectured (or suspected) nonperiodic binary sequences where there are no more than two consecutive 0's or 1's include: A083035, A285305, A190843, A286059, A288213, A288551, A288473, A176405, A188321, A188398, A191162, A272170, A197879, A078588, A272532, A273129, A074937, A188297, A289128. Others?

Cf. A083035, A285305, A190843, A286059, A288213, A288551, A288473, A176405, A188321, A188398, A191162, A272170, A197879, A078588, A272532, A273129,A074937, A188297, A289128.

c[n_]:=c[n]=(c[n-1]-2c[n-2]+c[n-3])/n;
c[1] = 0; c[2] = 0; c[3] = 1;
Table[Ceiling@c[j],{j,1,2^7}]

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A272532 Single bit representation of the sum of two sinusoids with periods 2 and 2*sqrt(2).

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A176409 Multiples of 3 or 7.

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A335617 Let c(1) = c(2) = 0, c(3) = 1, and c(n + 3) = (c(n) - 2*c(n + 1) + c(n + 2))/n, then a(n) = ceiling (c(n)).

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