A244845 Binary representation of 4^n - 2^(n+1) - 1.

111, 101111, 11011111, 1110111111, 111101111111, 11111011111111, 1111110111111111, 111111101111111111, 11111111011111111111, 1111111110111111111111, 111111111101111111111111, 11111111111011111111111111, 1111111111110111111111111111

Offset: 2

Author

Colin Barker, Jul 07 2014

Keywords

• nonn
• base
• easy

Examples

			a(3) is 101111 because A093112(3) = 47 which is 101111 in base 2.
		

Links

• Colin Barker, Table of n, a(n) for n = 2..500
• Wikipedia, Carol number
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A093112.

Programs

Mathematica

Table[FromDigits[IntegerDigits[4^n-2^(n+1)-1,2]],{n,2,15}] (* Harvey P. Dale, Oct 03 2016 *)

PARI

vector(100, n, (100^(n+1)-9*10^(2+n)-1)/9)

PARI

Vec(x^2*(89000*x^2-88790*x-111)/((x-1)*(10*x-1)*(100*x-1)) + O(x^100))

PARI

a(n) = subst(Pol(binary(4^n-2^(n+1)-1)), x, 10); \\ Michel Marcus, Jul 08 2014

Formula

a(n) = 111*a(n-1)-1110*a(n-2)+1000*a(n-3).

a(n) = (-1-9*10^(1+n)+100^n)/9.

G.f.: x^2*(89000*x^2-88790*x-111) / ((x-1)*(10*x-1)*(100*x-1)).