A090590 - OEIS

%I #37 Dec 22 2025 01:56:15

%S 1,-6,-20,8,176,288,-832,-3968,-1280,29184,68608,-96256,-741376,

%T -712704,4505600,14712832,-6619136,-130940928,-208928768,629669888,

%U 2930769920,824180736,-21797797888,-50189041664,74004299776,549520932864,507007467520,-3382152527872

%N (1,1) 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 Also real part of (1 +- i*sqrt(7))^n. - _Bruno Berselli_, Jun 24-25 2011

%H Vincenzo Librandi, <a href="/A090590/b090590.txt">Table of n, a(n) for n = 1..202</a>

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

%F G.f.: x*(1-8*x)/(1-2*x+8*x^2). - _T. D. Noe_, Dec 11 2006

%F From _Bruno Berselli_, Jun 24-25 2011: (Start)

%F a(n) = (1/2)*((1+i*sqrt(7))^n + (1-i*sqrt(7))^n), where i=sqrt(-1).

%F a(n) = cos(n*arctan(sqrt(7)))*sqrt(8)^n.

%F a(n) = 2*a(n-1) - 8*a(n-2) (n > 2). (End)

%F E.g.f.: exp(x) * (cos(sqrt(7)*x) - sqrt(7)*sin(sqrt(7)*x)). - _Amiram Eldar_, Dec 22 2025

%p 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 LinearRecurrence[{2,-8},{1,-6},30] (* _Harvey P. Dale_, Mar 30 2019 *)

%o (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 (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 (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 Cf. A087621, A090591.

%K sign,easy

%O 1,2

%A _Simone Severini_, Dec 04 2003

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

%E More terms from _Bruno Berselli_, Jun 24 2011