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A100401 Digital root of 3^n.

%I A100401 #66 Mar 20 2025 12:11:05
%S A100401 1,3,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,
%T A100401 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,
%U A100401 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A100401 Digital root of 3^n.
%C A100401 This sequence also gives the digital root of 12^n, 21^n, 30^n, 39^n, 48^n, 57^n, ... (any k^n where k is congruent to 3 mod 9). - _Timothy L. Tiffin_, Dec 02 2023
%H A100401 <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).
%F A100401 a(n) = 3^n mod 18. - _Zerinvary Lajos_, Nov 25 2009
%F A100401 From _Timothy L. Tiffin_, Nov 30 2023: (Start)
%F A100401 a(n) = 9 for n >= 2.
%F A100401 G.f.: (1+2x+6x^2)/(1-x).
%F A100401 a(n) = A100403(n) for n <> 1. (End)
%F A100401 a(n) = A010888(A000244(n)). - _Michel Marcus_, Dec 01 2023
%F A100401 a(n) = A010888(A001021(n)) = A010888(A009965(n)) = A010888(A009974(n)) = A010888(A009983(n)) = A010888(A009992(n)) = A010888(A225374(n)). - _Timothy L. Tiffin_, Dec 02 2023
%F A100401 E.g.f.: 9*exp(x) - 6*x - 8. - _Elmo R. Oliveira_, Aug 08 2024
%F A100401 a(n) = A007953(3*a(n-1)) = A010888(3*a(n-1)). - _Stefano Spezia_, Mar 20 2025
%e A100401 For n=14, the digits of 3^14 = 4782969 sum to 45, whose digits sum to 9. So, a(14) = 9.
%t A100401 Table[PowerMod[3, n, 18], {n, 0, 100}] (* _Timothy L. Tiffin_, Dec 03 2023 *)
%t A100401 PadRight[{1,3}, 100, 9] (* _Timothy L. Tiffin_, Dec 03 2023 *)
%o A100401 (Sage) [power_mod(3,n,18) for n in range(105)] # _Zerinvary Lajos_, Nov 25 2009
%o A100401 (PARI) a(n) = if( n<2, [1,3][n+1], 9); \\ _Joerg Arndt_, Dec 03 2023
%Y A100401 Cf. A000244, A001021, A007953, A009965, A009974, A009983, A009992, A010888, A100403, A225374.
%K A100401 easy,nonn,base
%O A100401 0,2
%A A100401 _Cino Hilliard_, Dec 30 2004