|
|
A010873
|
|
a(n) = n mod 4.
|
|
123
|
|
|
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;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
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
|
|
|
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))).
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) = 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
|
|
MAPLE
|
|
|
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 *)
|
|
PROG
|
(Haskell)
a010873 n = (`mod` 4)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
First to third formulas re-edited for better readability by Hieronymus Fischer, Dec 05 2011
|
|
STATUS
|
approved
|
|
|
|