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

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

Offset

2,1

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).

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)).

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A093112.

Sequence in context: A262641 A145698 A362920 * A152775 A362118 A242646

Adjacent sequences: A244842 A244843 A244844 * A244846 A244847 A244848

Keyword

nonn,base,easy

Author

Colin Barker, Jul 07 2014

Status

approved