A010036 Sum of 2^n, ..., 2^(n+1) - 1.

1, 5, 22, 92, 376, 1520, 6112, 24512, 98176, 392960, 1572352, 6290432, 25163776, 100659200, 402644992, 1610596352, 6442418176, 25769738240, 103079084032, 412316598272, 1649266917376, 6597068718080, 26388276969472, 105553112072192, 422212456677376

Offset: 0

Author

Steve King (ITTTUCSON(AT)aol.com)

Keywords

• nonn
• easy

Comments

a(n) = sum of next 2^n natural numbers. - Amarnath Murthy, Apr 17 2003

Sum of all proper binary numbers with n digits (i.e. those not beginning with 0). Cf. A101291 Sum of all numbers with n digits [base 10]. - Jonathan Vos Post, Sep 07 2006

a(n)/2^n gives the average eccentricity of the graphs of the Chinese rings puzzle with n+1 rings (also known as baguenaudier). - Daniele Parisse, Jun 02 2008

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 59. Book's website
• Andreas M. Hinz and Daniele Parisse, The Average Eccentricity of Sierpinski Graphs, Graphs and Combinatorics, 2011.
• Index entries for linear recurrences with constant coefficients, signature (6, -8).

Crossrefs

Cf. A010036.

Partial sums are in A006516, A006095.

Programs

Magma

[ &+[ k: k in [2^n..2^(n+1)-1] ]: n in [0..21] ]; // Klaus Brockhaus, Nov 27 2009

Magma

[2^n *(2^n+(2^(n+1)-1))/2: n in [0..25]]; // Vincenzo Librandi, Sep 11 2015

Maple

f:= n-> 3*2^(2*n-1)-2^(n-1): seq(f(n), n=0..30);

Mathematica

Table[2^n (2^n+(2^(n+1)-1))/2,{n,0,25}] (* or *) LinearRecurrence[{6,-8},{1,5},30] (* Harvey P. Dale, Jan 23 2012 *)
With[{nn=30},Total/@TakeList[Range[2^(nn+1)-1],2^Range[0,nn]]] (* Harvey P. Dale, May 26 2024 *)

PARI

a(n)=3<<(2*n-1)-1<<(n-1) \\ Charles R Greathouse IV, Jul 02 2013

Formula

a(n+1) = 4*a(n) + 2^n with a(0) = 1 (with a(0)=0, see A006516). a(n) = 2^(n-1)*A055010(n). - Philippe Deléham, Feb 20 2004

a(n) = 3*2^(2*n-1) - 2^(n-1). - Daniele Parisse, Jun 10 2007

From Klaus Brockhaus, Nov 27 2009: (Start)

a(n) = 6*a(n-1)-8*a(n-2) for n > 1; a(0) = 1, a(1) = 5.

G.f.: (1-x)/((1-2*x)*(1-4*x)). (End)

a(n) = Sum_{k, 0<=k<=n} A125185(n,k)*2^k. - Philippe Deléham, Feb 26 2012

a(n) = A006516(n+1)-A006516(n). - R. J. Mathar, Mar 06 2017