A067895 Write 2^n, 2^n+1, 2^n+2, ..., 2^(n+1)-1 in binary and add as if they were decimal numbers.

1, 21, 422, 8444, 168888, 3377776, 67555552, 1351111104, 27022222208, 540444444416, 10808888888832, 216177777777664, 4323555555555328, 86471111111110656, 1729422222222221312, 34588444444444442624, 691768888888888885248, 13835377777777777770496, 276707555555555555540992

Offset: 0

Author

N. J. A. Sloane, May 15 2003

Keywords

• nonn
• base
• easy

Examples

			2^2 to 2^3-1 = 4 through 7 = 100, 101, 110 and 111 in binary and when summed = 422.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (22,-40).

Crossrefs

Cf. A067894.

Programs

Mathematica

Table[Total[FromDigits[IntegerDigits[#,2]]&/@(Range[2^n,2^(n+1)-1])],{n,0,20}] (* Harvey P. Dale, May 20 2012 *)

PARI

a(n)=if(n<0,0,2^(n-1)*(19*10^n-1)/9)

Formula

G.f.: (1 - x)/(1 - 22x + 40*x^2).

a(n) = 2^(n-1)*(19*10^n - 1)/9.

a(n) = 22*a(n-1) - 40*a(n-2).

E.g.f.: exp(2*x)*(19*exp(18*x) - 1)/18. - Stefano Spezia, Apr 03 2023