A100401 Digital root of 3^n.

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, 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, 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

Offset: 0

Author

Cino Hilliard, Dec 30 2004

Keywords

• easy
• nonn
• base

Comments

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

Examples

			For n=14, the digits of 3^14 = 4782969 sum to 45, whose digits sum to 9. So, a(14) = 9.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1).

Crossrefs

Cf. A000244, A001021, A007953, A009965, A009974, A009983, A009992, A010888, A100403, A225374.

Programs

Mathematica

Table[PowerMod[3, n, 18], {n, 0, 100}] (* Timothy L. Tiffin, Dec 03 2023 *)
PadRight[{1,3}, 100, 9] (* Timothy L. Tiffin, Dec 03 2023 *)

PARI

a(n) = if( n<2, [1,3][n+1], 9); \\ Joerg Arndt, Dec 03 2023

Sage

[power_mod(3,n,18) for n in range(105)] # Zerinvary Lajos, Nov 25 2009

Formula

a(n) = 3^n mod 18. - Zerinvary Lajos, Nov 25 2009

From Timothy L. Tiffin, Nov 30 2023: (Start)

a(n) = 9 for n >= 2.

G.f.: (1+2x+6x^2)/(1-x).

a(n) = A100403(n) for n <> 1. (End)

a(n) = A010888(A000244(n)). - Michel Marcus, Dec 01 2023

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

E.g.f.: 9*exp(x) - 6*x - 8. - Elmo R. Oliveira, Aug 08 2024

a(n) = A007953(3*a(n-1)) = A010888(3*a(n-1)). - Stefano Spezia, Mar 20 2025