A090591 - OEIS

%I #36 Sep 08 2022 08:45:12

%S 1,2,-4,-24,-16,160,448,-384,-4352,-5632,23552,92160,-4096,-745472,

%T -1458176,3047424,17760256,11141120,-119799808,-328728576,300941312,

%U 3231711232,4055891968,-17741905920,-67930947584

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

%C (1,2) entry of powers of the orthogonal design shown below:

%C +1 +1 +1 +1 +1 +1 +1 +1

%C -1 +1 +1 -1 +1 -1 -1 +1

%C -1 -1 +1 +1 +1 +1 -1 -1

%C -1 +1 -1 +1 +1 -1 +1 -1

%C -1 -1 -1 -1 +1 +1 +1 +1

%C -1 +1 -1 +1 -1 +1 -1 +1

%C -1 +1 +1 -1 -1 +1 +1 -1

%C -1 -1 +1 +1 -1 -1 +1 +1

%C 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 G. C. Greubel, <a href="/A090591/b090591.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-8).

%F Binomial transform of (1+x)/(1+7*x^2).

%F 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 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 LinearRecurrence[{2,-8}, {1,2}, 30] (* _G. C. Greubel_, Oct 22 2018 *)

%t CoefficientList[Series[1/(1-2x+8x^2),{x,0,60}],x] (* _Harvey P. Dale_, Jan 17 2021 *)

%o (Sage) [lucas_number1(n,2,8) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 23 2009

%o (PARI) x='x+O('x^30); Vec(1/(1 - 2*x + 8*x^2)) \\ _G. C. Greubel_, Oct 22 2018

%o (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 (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 Cf. A087621, A090590.

%K sign

%O 0,2

%A _Simone Severini_, Dec 04 2003

%E Formulae from _Paul Barry_, Dec 05 2003

%E Corrected by _T. D. Noe_, Dec 11 2006