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A090590 (1,1) entry of powers of the orthogonal design shown below.

%I A090590 #32 Sep 08 2022 08:45:12
%S A090590 1,-6,-20,8,176,288,-832,-3968,-1280,29184,68608,-96256,-741376,
%T A090590 -712704,4505600,14712832,-6619136,-130940928,-208928768,629669888,
%U A090590 2930769920,824180736,-21797797888,-50189041664,74004299776,549520932864
%N A090590 (1,1) entry of powers of the orthogonal design shown below.
%C A090590 +1 +1 +1 +1 +1 +1 +1 +1
%C A090590 -1 +1 +1 -1 +1 -1 -1 +1
%C A090590 -1 -1 +1 +1 +1 +1 -1 -1
%C A090590 -1 +1 -1 +1 +1 -1 +1 -1
%C A090590 -1 -1 -1 -1 +1 +1 +1 +1
%C A090590 -1 +1 -1 +1 -1 +1 -1 +1
%C A090590 -1 +1 +1 -1 -1 +1 +1 -1
%C A090590 -1 -1 +1 +1 -1 -1 +1 +1
%C A090590 Also real part of (1 +- i*sqrt(7))^n. - _Bruno Berselli_, Jun 24-25 2011
%H A090590 Vincenzo Librandi, <a href="/A090590/b090590.txt">Table of n, a(n) for n = 1..202</a>
%H A090590 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-8).
%F A090590 G.f.: x*(1-8*x)/(1-2*x+8*x^2). - _T. D. Noe_, Dec 11 2006
%F A090590 From _Bruno Berselli_, Jun 24-25 2011: (Start)
%F A090590 a(n) = (1/2)*((1+i*sqrt(7))^n + (1-i*sqrt(7))^n), where i=sqrt(-1).
%F A090590 a(n) = cos(n*arctan(sqrt(7)))*sqrt(8)^n.
%F A090590 a(n) = 2*a(n-1) - 8*a(n-2) (n > 2). (End)
%p A090590 a := proc(n) option remember: if(n=1)then return 1:elif(n=2)then return -6:fi: return 2*a(n-1)-8*a(n-2): end: seq(a(n),n=1..26); # _Nathaniel Johnston_, Jun 25 2011
%t A090590 LinearRecurrence[{2,-8},{1,-6},30] (* _Harvey P. Dale_, Mar 30 2019 *)
%o A090590 (Magma) m:=27; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-8*x)/(1-2*x+8*x^2))); // _Bruno Berselli_, Jun 24-25 2011
%o A090590 (Maxima) makelist(expand(((1+sqrt(-1)*sqrt(7))^n+(1-sqrt(-1)*sqrt(7))^n)/2),n,1,26); /* _Bruno Berselli_, Jun 24-25 2011 */
%o A090590 (PARI) a=vector(26); a[1]=1; a[2]=-6; for(i=3, #a, a[i]=2*a[i-1]-8*a[i-2]); a \\ _Bruno Berselli_, Jun 24-25 2011
%Y A090590 Cf. A087621, A090591.
%K A090590 sign,easy
%O A090590 1,2
%A A090590 _Simone Severini_, Dec 04 2003
%E A090590 Corrected by _T. D. Noe_, Dec 11 2006
%E A090590 More terms from _Bruno Berselli_, Jun 24 2011