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, 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, 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
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 *)
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
KEYWORD
nonn,easy
AUTHOR
Steve King (ITTTUCSON(AT)aol.com)
STATUS
approved