A122803 Powers of -2: a(n) = (-2)^n.

1, -2, 4, -8, 16, -32, 64, -128, 256, -512, 1024, -2048, 4096, -8192, 16384, -32768, 65536, -131072, 262144, -524288, 1048576, -2097152, 4194304, -8388608, 16777216, -33554432, 67108864, -134217728, 268435456, -536870912, 1073741824, -2147483648, 4294967296, -8589934592, 17179869184

Offset: 0

Author

Franklin T. Adams-Watters, Sep 11 2006

Keywords

• easy
• sign

Comments

The number -2 can be used as a base of numeration (see the Weisstein link). - Alonso del Arte, Mar 30 2014

Contribution from M. F. Hasler, Oct 21 2014: (Start)

This is the inverse binomial transform of A033999 = n->(-1)^n, and the binomial transform of A033999*A000244 = n->(-3)^n, see also A141413.

Prefixed with one 0, i.e., (0,1,-2,4,...) = -A033999*A131577, it is the binomial transform of (0, 1, -4, 13, -40, 121,...) = -A033999*A003462, and inverse binomial transform of (0,1,0,1,0,1,...) = A000035.

Prefixed with two 0's, i.e., (0,0,1,-2,4,-8,...), it is the binomial transform of (0,0,1,-5,18,-58,179,-543,...) (cf. A000340) and inverse binomial transform of (0,0,1,1,2,2,3,3,...) = A004526. (End)

Prefixed with three 0's, this is the inverse binomial difference of (0, 0, 0, 1, 2, 4, 6, 9, 12, 16,...) = concat(0, A002620), which has as successive differences (0, 0, 1, 1, 2, 2,...) = A004526, then (0, 1, 0, 1,...) = A000035, then (1, -1, 1, -1,...) = A033999, and then (-2)^k*A033999 with k=1,2,3,... - Paul Curtz, Oct 16 2014, edited by M. F. Hasler, Oct 21 2014

Stirling-Bernoulli transform of triangular numbers: 1, 3, 6, 10, 15, 21, 28, ... - Philippe Deléham, May 25 2015

Links

• Franklin T. Adams-Watters, Table of n, (-2)^n for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Eric Weisstein's World of Mathematics, Negabinary.
• Index entries for linear recurrences with constant coefficients, signature (-2).

Crossrefs

Cf. A000079 (powers of 2), A004526, A005351, A005352, A011782, A034008, A131577, A171449, A248800.

Programs

Magma

[(-2)^n: n in [0..60]]; // Vincenzo Librandi, Oct 22 2014

Maple

A122803:=n->(-2)^n; seq(A122803(n), n=0..50); # Wesley Ivan Hurt, Mar 30 2014

Mathematica

Table[(-2)^n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Feb 15 2011 *)

PARI

a(n)=(-2)^n \\ Charles R Greathouse IV, Sep 24 2015

Python

def A122803(n): return -(1<Chai Wah Wu, Nov 18 2022

Formula

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

a(n) = -2*a(n-1), n > 0; a(0) = 1. G.f.: 1/(1+2x). - Philippe Deléham, Nov 19 2008

Sum_{n >= 0} 1/a(n) = 2/3. - Jaume Oliver Lafont, Mar 01 2009

E.g.f.: 1/exp(2*x). - Arkadiusz Wesolowski, Aug 13 2012

a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A030195(n+1). - R. J. Mathar, Oct 15 2012

G.f.: 1/(1+2x). A122803 = A033999 * A000079. - M. F. Hasler, Oct 21 2014

a(n) = Sum_{k = 0..n} A163626(n,k)*A000217(k+1). - Philippe Deléham, May 25 2015