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A090591 Expansion of g.f.: 1/(1 - 2*x + 8*x^2).

%I A090591 #36 Sep 08 2022 08:45:12
%S A090591 1,2,-4,-24,-16,160,448,-384,-4352,-5632,23552,92160,-4096,-745472,
%T A090591 -1458176,3047424,17760256,11141120,-119799808,-328728576,300941312,
%U A090591 3231711232,4055891968,-17741905920,-67930947584
%N A090591 Expansion of g.f.: 1/(1 - 2*x + 8*x^2).
%C A090591 (1,2) entry of powers of the orthogonal design shown below:
%C A090591 +1 +1 +1 +1 +1 +1 +1 +1
%C A090591 -1 +1 +1 -1 +1 -1 -1 +1
%C A090591 -1 -1 +1 +1 +1 +1 -1 -1
%C A090591 -1 +1 -1 +1 +1 -1 +1 -1
%C A090591 -1 -1 -1 -1 +1 +1 +1 +1
%C A090591 -1 +1 -1 +1 -1 +1 -1 +1
%C A090591 -1 +1 +1 -1 -1 +1 +1 -1
%C A090591 -1 -1 +1 +1 -1 -1 +1 +1
%C A090591 Pisano period lengths: 1, 1, 8, 1, 24, 8, 7, 1, 24, 24, 10, 8, 56, 7, 24, 1, 144, 24, 120, 24, ... - _R. J. Mathar_, Aug 10 2012
%H A090591 G. C. Greubel, <a href="/A090591/b090591.txt">Table of n, a(n) for n = 0..1000</a>
%H A090591 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-8).
%F A090591 Binomial transform of (1+x)/(1+7*x^2).
%F A090591 a(0)=1, a(1)=2, a(n) = 2*a(n-1) - 8*a(n-2) for n>1. - _Philippe Deléham_, Sep 19 2009
%p A090591 seq(coeff(series(1/(1-2*x+8*x^2),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Oct 23 2018
%t A090591 LinearRecurrence[{2,-8}, {1,2}, 30] (* _G. C. Greubel_, Oct 22 2018 *)
%t A090591 CoefficientList[Series[1/(1-2x+8x^2),{x,0,60}],x] (* _Harvey P. Dale_, Jan 17 2021 *)
%o A090591 (Sage) [lucas_number1(n,2,8) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009
%o A090591 (PARI) x='x+O('x^30); Vec(1/(1 - 2*x + 8*x^2)) \\ _G. C. Greubel_, Oct 22 2018
%o A090591 (Magma) [n le 2 select n else 2*Self(n-1) - 8*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Oct 22 2018
%o A090591 (GAP) a:=[1,2];; for n in [3..30] do a[n]:=2*a[n-1]-8*a[n-2]; od; a; # _Muniru A Asiru_, Oct 23 2018
%Y A090591 Cf. A087621, A090590.
%K A090591 sign
%O A090591 0,2
%A A090591 _Simone Severini_, Dec 04 2003
%E A090591 Formulae from _Paul Barry_, Dec 05 2003
%E A090591 Corrected by _T. D. Noe_, Dec 11 2006