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A010873 Simple periodic sequence. 42
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)
OFFSET

0,3

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

LINKS

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

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]

MAPLE

seq(chrem( [n, n], [1, 4] ), n=0..80); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2009]

PROG

(PARI) a(n)=n%4 \\ Charles R Greathouse IV, Dec 05 2011

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

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane (njas(AT)research.att.com).

EXTENSIONS

Second to fourth formulas re-edited for better readability by Hieronymus Fischer, Dec 05 2011

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Last modified February 15 09:26 EST 2012. Contains 205753 sequences.