OFFSET
0,2
COMMENTS
Digital root of 2^n.
A regular version of Pitoun's sequence: a(n) = A029898(n+1).
Also obtained from permutations of A141425, A020806, A070366, A153110, A153990, A154127, A154687, or A154815.
This sequence and its (again period 6) repeated differences produce the table:
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, ...
1, 2, 4, -1, -2, -4, 1, 2, 4, -1, -2, ...
1, 2, -5, -1, -2, 5, 1, 2, -5, -1, -2, ...
1, -7, 4, -1, 7, -4, 1, -7, 4, -1, 7, ...
-8, 11, -5, 8,-11, 5, -8, 11, -5, 8,-11, ...
19,-16, 13,-19, 16,-13, 19,-16, 13,-19, 16, ...
-35, 29,-32, 35,-29, 32,-35, 29,-32, 35,-29, ...
64,-61, 67,-64, 61,-67, 64,-61, 67,-64, 61, ...
If each entry of this table is read modulo 9 we obtain the very regular table:
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, ...
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, ...
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, ...
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, ...
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, ...
1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, ...
Also the decimal expansion of the constant 125/1001. - R. J. Mathar, Jan 23 2009
Digital root of the powers of any number congruent to 2 mod 9. - Alonso del Arte, Jan 26 2014
REFERENCES
Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.
LINKS
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).
FORMULA
a(n) + a(n+3) = 9 = A010734(n).
G.f.: (1+x+2x^2+5x^3)/((1-x)(1+x)(1-x+x^2)). - R. J. Mathar, Jan 23 2009
a(n) = A082365(n) mod 9. - Paul Curtz, Mar 31 2009
a(n) = -1/2*cos(Pi*n) - 3*cos(1/3*Pi*n) - 3^(1/2)*sin(1/3*Pi*n) + 9/2. - Leonid Bedratyuk, May 13 2012
From Wesley Ivan Hurt, Apr 20 2015: (Start)
a(n) = a(n-6) for n>5.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (2+3*(n-1 mod 3))*(n mod 2) + (1+3*(-n mod 3))*(n-1 mod 2). (End)
a(n) = 2^n mod 9. - Nikita Sadkov, Oct 06 2018
MAPLE
seq(op([1, 2, 4, 8, 7, 5]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016
MATHEMATICA
Flatten[Table[{1, 2, 4, 8, 7, 5}, {20}]] (* Paul Curtz, Dec 19 2008 *)
Table[Mod[2^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)
PROG
(PARI) a(n)=lift(Mod(2, 9)^n) \\ Charles R Greathouse IV, Apr 21 2015
(Magma) &cat [[1, 2, 4, 8, 7, 5]^^30]; // Wesley Ivan Hurt, Jul 05 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Dec 19 2008
EXTENSIONS
Edited by R. J. Mathar, Apr 09 2009
STATUS
approved