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A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

%I A153130 #75 Mar 20 2025 12:09:50
%S A153130 1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,
%T A153130 7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,
%U A153130 4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5
%N A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].
%C A153130 Digital root of 2^n.
%C A153130 A regular version of Pitoun's sequence: a(n) = A029898(n+1).
%C A153130 Also obtained from permutations of A141425, A020806, A070366, A153110, A153990, A154127, A154687, or A154815.
%C A153130 This sequence and its (again period 6) repeated differences produce the table:
%C A153130     1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
%C A153130     1,  2,  4, -1, -2, -4,  1,  2,  4, -1, -2, ...
%C A153130     1,  2, -5, -1, -2,  5,  1,  2, -5, -1, -2, ...
%C A153130     1, -7,  4, -1,  7, -4,  1, -7,  4, -1,  7, ...
%C A153130    -8, 11, -5,  8,-11,  5, -8, 11, -5,  8,-11, ...
%C A153130    19,-16, 13,-19, 16,-13, 19,-16, 13,-19, 16, ...
%C A153130   -35, 29,-32, 35,-29, 32,-35, 29,-32, 35,-29, ...
%C A153130    64,-61, 67,-64, 61,-67, 64,-61, 67,-64, 61, ...
%C A153130 If each entry of this table is read modulo 9 we obtain the very regular table:
%C A153130     1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
%C A153130     1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
%C A153130     1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
%C A153130     1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
%C A153130     1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
%C A153130     1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
%C A153130 Also the decimal expansion of the constant 125/1001. - _R. J. Mathar_, Jan 23 2009
%C A153130 Digital root of the powers of any number congruent to 2 mod 9. - _Alonso del Arte_, Jan 26 2014
%D A153130 Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.
%H A153130 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1).
%F A153130 a(n) + a(n+3) = 9 = A010734(n).
%F A153130 G.f.: (1+x+2x^2+5x^3)/((1-x)(1+x)(1-x+x^2)). - _R. J. Mathar_, Jan 23 2009
%F A153130 a(n) = A082365(n) mod 9. - _Paul Curtz_, Mar 31 2009
%F A153130 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
%F A153130 a(n) = A010888(A004000(n+1)). - _Ivan N. Ianakiev_, Nov 27 2014
%F A153130 From _Wesley Ivan Hurt_, Apr 20 2015: (Start)
%F A153130 a(n) = a(n-6) for n>5.
%F A153130 a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
%F A153130 a(n) = (2+3*(n-1 mod 3))*(n mod 2) + (1+3*(-n mod 3))*(n-1 mod 2). (End)
%F A153130 a(n) = 2^n mod 9. - _Nikita Sadkov_, Oct 06 2018
%F A153130 From _Stefano Spezia_, Mar 20 2025: (Start)
%F A153130 E.g.f.: 4*cosh(x) - exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 5*sinh(x).
%F A153130 a(n) = A007953(2*a(n-1)) = A010888(2*a(n-1)). (End)
%p A153130 seq(op([1, 2, 4, 8, 7, 5]), n=0..40); # _Wesley Ivan Hurt_, Jul 05 2016
%t A153130 Flatten[Table[{1, 2, 4, 8, 7, 5}, {20}]] (* _Paul Curtz_, Dec 19 2008 *)
%t A153130 Table[Mod[2^n, 9], {n, 0, 99}] (* _Alonso del Arte_, Jan 26 2014 *)
%o A153130 (PARI) a(n)=lift(Mod(2,9)^n) \\ _Charles R Greathouse IV_, Apr 21 2015
%o A153130 (Magma) &cat [[1, 2, 4, 8, 7, 5]^^30]; // _Wesley Ivan Hurt_, Jul 05 2016
%Y A153130 Cf. A004000, A007953, A010734, A010888, A029898, A030132, A082365, A145389, A189510.
%Y A153130 Cf. digital roots of powers of c mod 9: c = 4, A100402; c = 5, A070366; c = 7, A070403; c = 8, A010689.
%Y A153130 Cf. A020806, A070366, A141425, A153110, A153990, A154127, A154687, A154815.
%K A153130 nonn,easy
%O A153130 0,2
%A A153130 _Paul Curtz_, Dec 19 2008
%E A153130 Edited by _R. J. Mathar_, Apr 09 2009