|
| |
|
|
A010877
|
|
Simple periodic sequence.
|
|
21
| |
|
|
0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| The rightmost digit in the base-8 representation of n. Also, the equivalent value of the three rightmost digits in the base-2 representation of n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 12 2007
|
|
|
LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).
|
|
|
FORMULA
| a(n)=n mod 8.
Complex representation: a(n)=1/8*(1-r^n)*sum{1<=k<8, k*product{1<=m<8,m<>k, (1-r^(n-m))}} where r=exp(pi/4*i)=(1+i)*sqrt(2)/2 and i=sqrt(-1).
Trigonometric representation: a(n)=256*(sin(n*pi/8))^2*sum{1<=k<8, k*product{1<=m<8,m<>k, (sin((n-m)*pi/8))^2}}.
G.f.: g(x)=(sum{1<=k<8, k*x^k})/(1-x^8).
Also: g(x)=x(7x^8-8x^7+1)/((1-x^8)(1-x)^2). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 31 2007
a(n)=n mod 2+2*(floor(n/2)mod 4)=A000035(n)+2*A010873(A004526(n)).
a(n)=n mod 4+4*(floor(n/4)mod 2)=A010873(n)+4*A000035(A002265(n)).
a(n)=n mod 2+2*(floor(n/2)mod 2)+4*(floor(n/4)mod 2)=A000035(n)+2*A000035(A004526(n)))+4*A000035(A002265(n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 12 2007
a(n)=(1/2)*(7-(-1)^n-2*(-1)^(b/4)-4*(-1)^((b-2+2*(-1)^(b/4))/8)) where b=2n-1+(-1)^n. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 12 2007
General formula for period 2^k: a(n)=(1/2)*(2^k-1-sum{0<=j<k, 2^j*(-1)^p(j,n)}) where p(j,n) is defined recursively by p(0,n)=n, p(j,n)=(1/4)*(2*p(j-1,n)-1+(-1)^p(j-1,n)). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jun 14 2007
|
|
|
MATHEMATICA
| Table[Mod[n, 8], {n, 0, 120}] (* From Harvey P. Dale, Apr 21 2011 *)
|
|
|
PROG
| (PARI) vector(100, i, i)%8 \\ Charles R Greathouse IV, Jul 16, 2011
|
|
|
CROSSREFS
| Partial sums: A130486. Other related sequences A130481, A130482, A130483, A130484, A130485.
Cf. A010887, A130909, A010873, A130909.
Sequence in context: A037850 A037886 A031045 * A195831 A004183 A031035
Adjacent sequences: A010874 A010875 A010876 * A010878 A010879 A010880
|
|
|
KEYWORD
| nonn,easy
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Formula section re-edited for better readability by Hieronymus Fischer
|
| |
|
|
