A010873 a(n) = n mod 4.

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

Offset

0,3

Comments

Complement of A002265, since 4*A002265(n)+a(n) = n. - Hieronymus Fischer, 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, Jun 11 2007

Periodic sequences of this type can be also calculated by a(n) = floor(q/(p^m-1)*p^n) mod p, where q is the number representing the periodic digit pattern and m is the period length. p and q can be calculated as follows: Let D be the array representing the number pattern to be repeated, m = size of D, max = maximum value of elements in D. Than p := max + 1 and q := p^m*sum_{i=1..m} D(i)/p^i. Example: D = (0, 1, 2, 3), p = 4 and q = 57 for this sequence. - Hieronymus Fischer, Jan 04 2013

Links

Antti Karttunen, Table of n, a(n) for n = 0..65536

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

Formula

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.: (3x^3+2x^2+x)/(1-x^4). - Hieronymus Fischer, May 29 2007

From Hieronymus Fischer, Jun 11 2007: (Start)

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.

a(n) = n mod 2+2*(floor(n/2)mod 2) = A000035(n)+2*A000035(A004526(n)). (End)

a(n) = 6 - a(n-1) - a(n-2) - a(n-3) for n > 2. - Reinhard Zumkeller, Apr 13 2008

a(n) = 3/2 + cos((n+1)pi)/2 + sqrt(2)cos((2n+3)Pi/4) [Jaume Oliver Lafont, Dec 05 2008]

From Hieronymus Fischer, Jan 04 2013: (Start)

a(n) = floor(41/3333*10^(n+1)) mod 10.

a(n) = floor(19/85*4^(n+1)) mod 4. (End)

E.g.f.: 2*sinh(x) - sin(x) + cosh(x) - cos(x). - Stefano Spezia, Apr 20 2021

From Nicolas Bělohoubek, May 30 2024: (Start)

a(n) = (2*a(n-1)-1)*(2-a(n-2)) for n > 1.

a(n) = (2*a(n-1)^2+1)*(3-a(n-1))/3 for n > 0. (End)

Maple

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

Mathematica

nn=40; CoefficientList[Series[(x+2x^2+3x^3)/(1-x^4), {x, 0, nn}], x] (* Geoffrey Critzer, Jul 26 2013 *)
Table[Mod[n, 4], {n, 0, 100}] (* T. D. Noe, Jul 26 2013 *)
PadRight[{}, 120, {0, 1, 2, 3}] (* Harvey P. Dale, Mar 29 2018 *)

Prog

(PARI) a(n)=n%4 \\ Charles R Greathouse IV, Dec 05 2011
(Haskell)
a010873 n = (`mod` 4)
a010873_list = cycle [0..3]  -- Reinhard Zumkeller, Jun 05 2012
(Scheme) (define (A010873 n) (modulo n 4)) ;; Antti Karttunen, Nov 07 2017

Crossrefs

Partial sums: A130482. Other related sequences A130481, A130483, A130484, A130485.

Cf. A004526, A002264, A002265, A002266.

Cf. A000035, A010877.

Sequence in context: A319047 A276335 A189480 * A049804 A277904 A132387

Adjacent sequences: A010870 A010871 A010872 * A010874 A010875 A010876

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

First to third formulas re-edited for better readability by Hieronymus Fischer, Dec 05 2011

Incorrect g.f. removed by Georg Fischer, May 18 2019

Status

approved