A007758 a(n) = 2^n*n^2.

0, 2, 16, 72, 256, 800, 2304, 6272, 16384, 41472, 102400, 247808, 589824, 1384448, 3211264, 7372800, 16777216, 37879808, 84934656, 189267968, 419430400, 924844032, 2030043136, 4437573632, 9663676416, 20971520000, 45365592064, 97844723712, 210453397504

Offset: 0

Author

David J. Snook (ua532(AT)freenet.victoria.bc.ca)

Keywords

• nonn
• easy

Comments

"The traveling salesman problem can be solved in time O(n^2 2^n) (where n is the size of the network to visit)." [Wikipedia] - Jonathan Vos Post, Apr 10 2006

Satisfies Benford's law [Theodore P. Hill, Personal communication, Feb 06, 2017]. - N. J. A. Sloane, Feb 08 2017

References

• Arno Berger and Theodore P. Hill. An Introduction to Benford's Law. Princeton University Press, 2015.
• Konrad Knopp, Theory and Application of Infinite Series, Dover, p. 269.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
• Wikipedia, Complexity.
• Index entries for linear recurrences with constant coefficients, signature (6,-12,8).
• Index entries for sequences related to Benford's law.

Crossrefs

Cf. A014477, A248917, A355234.

Programs

Magma

[2^n*n^2: n in [0..30]]; // Vincenzo Librandi, Oct 27 2011

Maple

seq(seq(k^n*n^k, k=2..2), n=0..25); and seq(2^n*n^2, n=0..25); # Zerinvary Lajos, Jul 01 2007

Mathematica

Table[n^2 * 2^n, {n, 0, 31}] (* Alonso del Arte, Oct 22 2014 *)
LinearRecurrence[{6,-12,8},{0,2,16},30] (* Harvey P. Dale, Jan 27 2017 *)

PARI

a(n)=n^2<Charles R Greathouse IV, Oct 28 2014

Formula

From Henry Bottomley, Jun 13 2001: (Start)

a(n) = 2*A014477(n-1).

G.f.: 2*x(1+2*x)/(1-2*x)^3.

Binomial transform of A002939.

Inverse binomial transform of A062189. (End)

Sum_{n>=1} 1/a(n) = Pi^2/12 - (1/2)*(log(2))^2. - Benoit Cloitre, Apr 05 2002

a(n) = Sum_{k=1..n} k*2^k. - Zerinvary Lajos, Oct 09 2006

E.g.f.: exp(2*x)*(2*x + 4*x^2). - Geoffrey Critzer, Aug 28 2013

Sum_{n>=1} (-1)^(n+1)/a(n) = -Li_2(-1/2) (A355234). - Amiram Eldar, Jun 28 2022