|
%I #31 Sep 08 2022 08:45:51
%S 1,1,-2,-8,-16,-16,32,128,256,256,-512,-2048,-4096,-4096,8192,32768,
%T 65536,65536,-131072,-524288,-1048576,-1048576,2097152,8388608,
%U 16777216,16777216,-33554432,-134217728,-268435456,-268435456
%N A (3/2,-1) Somos-4 sequence.
%C Hankel transform of A051286. a(n+2) = -(-1)^floor(n/4) * 2^A098181(n).
%H G. C. Greubel, <a href="/A174882/b174882.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,-16).
%F a(n) = ((3/2)*a(n-1)*a(n-3) - a(n-2)^2)/a(n-4), n>3.
%F a(-n) = a(n-1) / 2^(2*n - 1) for all n in Z. - _Michael Somos_, Jan 06 2011
%F 0 = a(n)*(+2*a(n+4)) + a(n+1)*(-3*a(n+3)) + a(n+2)*(+2*a(n+2)) for all n in Z. - _Michael Somos_, Sep 18 2014
%F a(n+4) = -16 * a(n) for all n in Z. - _Michael Somos_, Sep 02 2015
%F G.f.: -(2*x-1)*(4*x^2+3*x+1)/(1+16*x^4) . - _R. J. Mathar_, Aug 18 2017
%e G.f. = 1 + x - 2*x^2 - 8*x^3 - 16*x^4 - 16*x^5 + 32*x^6 + 128*x^7 + ...
%t a[ n_] := (-1)^Quotient[n + 2, 4] 2^(n - Mod[ Quotient[n + 1, 2], 2]); (* _Michael Somos_, Sep 18 2014 *)
%t CoefficientList[Series[(1-2*x)*(4*x^2+3*x+1)/(1+16*x^4), {x,0,50}], x] (* _G. C. Greubel_, Feb 21 2018 *)
%o (PARI) {a(n) = (-1)^((n+2) \ 4) * 2^(n - ((n+1) \ 2 % 2))}; /* _Michael Somos_, Jan 06 2011 */
%o (PARI) x='x+O('x^30); Vec((1-2*x)*(4*x^2+3*x+1)/(1+16*x^4)) \\ _G. C. Greubel_, Feb 21 2018
%o (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((1-2*x)*(4*x^2+3*x+1)/(1+16*x^4))) // _G. C. Greubel_, Feb 21 2018
%Y Cf. A051286, A098181.
%K easy,sign
%O 0,3
%A _Paul Barry_, Mar 31 2010
|
