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A010873
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Simple periodic sequence.
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42
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0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 2, 3, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Complement of A002265, since 4*A002265(n)+a(n)=n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
The rightmost digit in the base-4 representation of n. Also, the equivalent value of the two rightmost digits in the base-2 representation of n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 11 2007
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (0,0,0,1).
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FORMULA
| a(n) = n mod 4
a(n)=(1/2)*(3-(-1)^n-2*(-1)^floor(n/2)); also a(n)=(1/2)*(3-(-1)^n-2*(-1)^((2n-1+(-1)^n)/4))); also a(n)=(1/2)*(3-(-1)^n-2*sin(pi/4*(2n+1+(-1)^n))).
G.f.: g(x)=(3x^3+2x^2+x)/(1-x^4). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 29 2007
Trigonometric representation: a(n)=2^2*(sin(n*pi/4))^2*sum{1<=k<4, k*product{1<=m<4,m<>k, (sin((n-m)*pi/4))^2}}. Clearly, the squared terms may be replaced by their absolute values '|.|'.
Complex representation: a(n)=1/4*(1-r^n)*sum{1<=k<4, k*product{1<=m<4,m<>k, (1-r^(n-m))}} where r=exp(pi/2*i)=i=sqrt(-1). All these formulas can be easily adapted to represent any periodic sequence.
G.f.: also g(x)=x(4x^5-5x^4+1)/((1-x^4)(1-x)^2). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 01 2007
a(n)=n mod 2+2*(floor(n/2)mod 2)=A000035(n)+2*A000035(A004526(n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 11 2007
a(n) = 6 - a(n-1) - a(n-2) - a(n-3) for n > 2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 13 2008
a(n) = 3/2 + cos((n+1)pi)/2 + sqrt(2)cos((2n+3)pi/4) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Dec 05 2008]
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MAPLE
| seq(chrem( [n, n], [1, 4] ), n=0..80); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]
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PROG
| (PARI) a(n)=n%4 \\ Charles R Greathouse IV, Dec 05 2011
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CROSSREFS
| Partial sums: A130482. Other related sequences A130481, A130483, A130484, A130485.
Cf. A004526, A002264, A002265, A002266.
Sequence in context: A096799 A106728 A189480 * A049804 A132387 A124757
Adjacent sequences: A010870 A010871 A010872 * A010874 A010875 A010876
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Second to fourth formulas re-edited for better readability by Hieronymus Fischer, Dec 05 2011
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