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A010872
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a(n) = n mod 3.
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123
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0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2
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OFFSET
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0,3
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COMMENTS
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Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 12.
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LINKS
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FORMULA
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a(n) = n - 3*floor(n/3) = a(n-3).
G.f.: (2*x^2+x)/(1-x^3). - Mario Catalani (mario.catalani(AT)unito.it), Jan 08 2003
a(n) = 1 + (1-2*cos(2*Pi*(n-1)/3)) * sin(2*Pi*(n-1)/3)) / sqrt(3).
a(n) = (1-r^n)*(1+r^n/(1-r)) where r=exp(2*Pi*i/3)=(-1+sqrt(3)*i)/2 and i=sqrt(-1). [corrected by Guenther Schrack, Sep 23 2019] (End)
a(n) = (16/9)*((sin(Pi*(n-2)/3))^2+2*(sin(Pi*(n-1)/3))^2)*(sin(Pi*n/3))^2.
a(n) = (4/3)*(|sin(Pi*(n-2)/3)|+2*|sin(Pi*(n-1)/3)|)*|sin(Pi*n/3)|.
a(n) = (4/9)*((1-cos(2*Pi*(n-2)/3))+2*(1-cos(2*Pi*(n-1)/3)))*(1-cos(2*Pi*n/3)). (End)
a(n) = 1 - 0^((-1)^(n/3)-(-1)^n) + 0^((-1)^((n+1)/3)+(-1)^n).
a(n) = 1 + (-1)^((2*n+4)/3)/3 + (-1)^((-2*n-4)/3)/3 + 2*(-1)^((2*n+2)/3)/3 + 2*(-1)^((-2*n-2)/3)/3.
a(n) = 1 + 2*cos(Pi*(2*n+4)/3)/3 + 4*cos(Pi*(2*n+2)/3)/3. (End)
a(n) = (r^n*(r-1) - r^(2*n)*(r + 2) + 3)/3 where r = (-1 + sqrt(-3))/2. - Guenther Schrack, Sep 23 2019
E.g.f.: exp(x) - exp(-x/2)*(cos(sqrt(3)*x/2) + sin(sqrt(3)*x/2)/sqrt(3)). - Stefano Spezia, Mar 01 2020
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EXAMPLE
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G.f. = x + 2*x^2 + x^4 + 2*x^5 + x^7 + 2*x^8 + x^10 + 2*x^11 + x^13 + ...
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MAPLE
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MATHEMATICA
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Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1}, 1 -> {2, 0}, 2 -> {1, 2}})]}], {0}, 7] (* Robert G. Wilson v, Feb 28 2005 *)
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PROG
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(Haskell)
a010872 = (`mod` 3)
(PARI) x='x+O('x^200); concat(0, Vec((2*x^2+x)/(1-x^3))) \\ Altug Alkan, Mar 23 2016
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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