A067895 - OEIS

%I #11 Apr 03 2023 20:12:52

%S 1,21,422,8444,168888,3377776,67555552,1351111104,27022222208,

%T 540444444416,10808888888832,216177777777664,4323555555555328,

%U 86471111111110656,1729422222222221312,34588444444444442624,691768888888888885248,13835377777777777770496,276707555555555555540992

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

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (22,-40).

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

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

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

%F E.g.f.: exp(2*x)*(19*exp(18*x) - 1)/18. - _Stefano Spezia_, Apr 03 2023

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

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

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

%Y Cf. A067894.

%K nonn,base,easy

%O 0,2

%A _N. J. A. Sloane_, May 15 2003