cp's OEIS Frontend

A084102 Inverse binomial transform of A084101.

%I A084102 #16 Oct 14 2022 08:55:14
%S A084102 1,2,-2,0,4,-8,8,0,-16,32,-32,0,64,-128,128,0,-256,512,-512,0,1024,
%T A084102 -2048,2048,0,-4096,8192,-8192,0,16384,-32768,32768,0,-65536,131072,
%U A084102 -131072,0,262144,-524288,524288,0,-1048576,2097152,-2097152,0,4194304,-8388608,8388608,0,-16777216,33554432
%N A084102 Inverse binomial transform of A084101.
%C A084102 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
%H A084102 G. C. Greubel, <a href="/A084102/b084102.txt">Table of n, a(n) for n = 0..1000</a>
%H A084102 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence#Specific_names">Lucas sequence</a>
%H A084102 <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%H A084102 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,-2).
%F A084102 G.f.: (1+2*x)^2/(1+2*x+2*x^2). - _Paul D. Hanna_, Nov 05 2009
%F A084102 From _G. C. Greubel_, Oct 13 2022: (Start)
%F A084102 a(n) = 2*A009116(n-1), n >= 1, with a(0) = 1.
%F A084102 a(n) = Real part of ( 2*(-1-i)^(n-1) + 2*[n=0] ).
%F A084102 a(n) = 2*(-1)^n*(2*(1+i)^(n-5) + i*(1-i)^(n-3)), n >= 1, with a(0) = 1.
%F A084102 E.g.f.: 2 - exp(-x)*(cos(x) - sin(x)). (End)
%t A084102 LinearRecurrence[{-2,-2},{1,2,-2},50] (* _Harvey P. Dale_, Aug 09 2017 *)
%o A084102 (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
%o A084102 (SageMath)
%o A084102 b=BinaryRecurrenceSequence(-2,-2,2,-2)
%o A084102 def A084102(n): return 1 if (n==0) else b(n-1)
%o A084102 [A084102(n) for n in range(41)] # _G. C. Greubel_, Oct 13 2022
%Y A084102 Yet another variation on A009545.
%Y A084102 Cf. A009116, A084101.
%K A084102 sign,easy
%O A084102 0,2
%A A084102 _Paul Barry_, May 15 2003