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A010036 Sum of 2^n, ..., 2^(n+1) - 1. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 (daniele.parisse(AT)eads.com), 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).

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 (daniele.parisse(AT)t-online.de), 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

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 *)

PROG

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

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

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

CROSSREFS

Cf. A010036.

Partial sums are in A006516, A006095.

Sequence in context: A053297 A071715 A278472 * A127617 A095932 A000346

Adjacent sequences:  A010033 A010034 A010035 * A010037 A010038 A010039

KEYWORD

nonn,easy

AUTHOR

Steve King (ITTTUCSON(AT)aol.com)

STATUS

approved

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Last modified January 22 22:38 EST 2018. Contains 298093 sequences.