A084102 Inverse binomial transform of A084101.

1, 2, -2, 0, 4, -8, 8, 0, -16, 32, -32, 0, 64, -128, 128, 0, -256, 512, -512, 0, 1024, -2048, 2048, 0, -4096, 8192, -8192, 0, 16384, -32768, 32768, 0, -65536, 131072, -131072, 0, 262144, -524288, 524288, 0, -1048576, 2097152, -2097152, 0, 4194304, -8388608, 8388608, 0, -16777216, 33554432

Offset: 0

Author

Paul Barry, May 15 2003

Keywords

• sign
• easy

Comments

The sequence {2, -2, 0, 4, -8, 8, 0, -16, 32, -32, 0, 64, -128, 128, 0, ...} (without the leading 1) is the Lucas V(-2, 2) sequence. - R. J. Mathar, Jan 08 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Wikipedia, Lucas sequence
• Index entries for Lucas sequences
• Index entries for linear recurrences with constant coefficients, signature (-2,-2).

Crossrefs

Yet another variation on A009545.

Cf. A009116, A084101.

Programs

Magma

[1] cat [n le 2 select 2*(-1)^(n-1) else -2*(Self(n-1) +Self(n-2)): n in [1..40]]; // G. C. Greubel, Oct 13 2022

Mathematica

LinearRecurrence[{-2,-2},{1,2,-2},50] (* Harvey P. Dale, Aug 09 2017 *)

SageMath

b=BinaryRecurrenceSequence(-2,-2,2,-2)
def A084102(n): return 1 if (n==0) else b(n-1)
[A084102(n) for n in range(41)] # G. C. Greubel, Oct 13 2022

Formula

G.f.: (1+2*x)^2/(1+2*x+2*x^2). - Paul D. Hanna, Nov 05 2009

From G. C. Greubel, Oct 13 2022: (Start)

a(n) = 2*A009116(n-1), n >= 1, with a(0) = 1.

a(n) = Real part of ( 2*(-1-i)^(n-1) + 2*[n=0] ).

a(n) = 2*(-1)^n*(2*(1+i)^(n-5) + i*(1-i)^(n-3)), n >= 1, with a(0) = 1.

E.g.f.: 2 - exp(-x)*(cos(x) - sin(x)). (End)