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

0,2

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

Links

Table of n, a(n) for n=0..104.

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

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

Example

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

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 *)

Prog

(Sage) [power_mod(3, n, 18) for n in range(105)] # Zerinvary Lajos, Nov 25 2009
(PARI) a(n) = if( n<2, [1, 3][n+1], 9); \\ Joerg Arndt, Dec 03 2023

Crossrefs

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

Sequence in context: A188444 A201409 A111120 * A004166 A110759 A063750

Adjacent sequences: A100398 A100399 A100400 * A100402 A100403 A100404

Keyword

easy,nonn,base

Author

Cino Hilliard, Dec 30 2004

Status

approved