A094014 Expansion of (1-2*x)/(1-8*x^2).

1, -2, 8, -16, 64, -128, 512, -1024, 4096, -8192, 32768, -65536, 262144, -524288, 2097152, -4194304, 16777216, -33554432, 134217728, -268435456, 1073741824, -2147483648, 8589934592, -17179869184, 68719476736, -137438953472

Offset: 0

Author

Paul Barry, Apr 21 2004

Keywords

• easy
• sign

Comments

Second inverse binomial transform of A094013. Third inverse binomial transform of A000129(2n-1).

The unsigned sequence has g.f. (1+2*x)/(1-8*x^2) and abs(a(n)) = 2^(3*n/2)*(1/2 + sqrt(2)/4 + (1/2 - sqrt(2)/4)*(-1)^n).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,8).

Crossrefs

Cf. A000129, A016116, A094013, A113836, A158020.

Programs

Magma

[n le 2 select (-2)^(n-1) else 8*Self(n-2): n in [1..41]]; // G. C. Greubel, Dec 04 2021

Mathematica

LinearRecurrence[{0,8}, {1,-2}, 40] (* G. C. Greubel, Dec 04 2021 *)

Sage

[(-2)^n*2^(n//2) for n in (0..40)] # G. C. Greubel, Dec 04 2021

Formula

a(n) = (2*sqrt(2))^n*(1/2 - sqrt(2)/4) + (-2*sqrt(2))^n*(1/2 + sqrt(2)/4).

a(n) = (-2)^n * A016116(n). - R. J. Mathar, Apr 28 2008

Abs(a(n)) = A113836(n+1) - A113836(n) for n > 0. - Reinhard Zumkeller, Feb 22 2010

a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011

a(n) = Sum_{k=0..n} A158020(n,k)*3^k. - Philippe Deléham, Dec 01 2011

E.g.f.: cosh(2*sqrt(2)*x) - (1/sqrt(2))*sinh(2*sqrt(2)*x). - G. C. Greubel, Dec 04 2021