A004663 Powers of 3 written in base 9.

1, 3, 10, 30, 100, 300, 1000, 3000, 10000, 30000, 100000, 300000, 1000000, 3000000, 10000000, 30000000, 100000000, 300000000, 1000000000, 3000000000, 10000000000, 30000000000, 100000000000, 300000000000, 1000000000000, 3000000000000, 10000000000000, 30000000000000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Links

• Robert Israel, Table of n, a(n) for n = 0..1997
• Index entries for linear recurrences with constant coefficients, signature (0,10).

Crossrefs

Cf. A026383, A016116.

Cf. A000244, A004656, A004658, A004659, ... : powers of 3 in base 10, 2, 4, 5, ...

Cf. A000079, A004642, ..., A004655: powers of 2 written in base 10, 2, 3, ..., 16.

Programs

Maple

seq(op([10^i,3*10^i]),i=0..100); # Robert Israel, Jun 25 2018

Mathematica

Table[FromDigits[IntegerDigits[3^n, 9]], {n, 0, 100}] (* G. C. Greubel, Oct 12 2018 *)

PARI

a(n)=3^bittest(n,0)*10^(n\2) \\ M. F. Hasler, Jun 25 2018

Formula

From Paul Barry, Jul 14 2004: (Start)

G.f.: (1 + 3*x)/(1 - 10*x^2);

a(n) = 2*a(n-1) + 3*a(n-2) + 10^floor((n-2)/2);

a(n) = Sum_{k=0..floor(n/2)} binomial(floor(n/2), k)*3^(n-2*k). (End)

a(n) = 3*a(n-1) + ((1 + (-1)^n)/2)*a(n-2) with a(0)=1, a(1)=3. - Taras Goy, Mar 20 2019

E.g.f.: cosh(sqrt(10)*x) + 3*sinh(sqrt(10)*x)/sqrt(10). - Stefano Spezia, Mar 31 2023