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A230368 A strong divisibility sequence associated with the algebraic integer 1 + i.

%I A230368 #25 Sep 03 2019 08:01:39
%S A230368 1,1,1,5,1,1,1,15,1,1,1,65,1,1,1,255,1,1,1,1025,1,1,1,4095,1,1,1,
%T A230368 16385,1,1,1,65535,1,1,1,262145,1,1,1,1048575,1,1,1,4194305,1,1,1,
%U A230368 16777215,1,1,1,67108865,1,1,1,268435455,1,1,1,1073741825
%N A230368 A strong divisibility sequence associated with the algebraic integer 1 + i.
%C A230368 Let alpha be an algebraic integer and define a sequence of integers a(n) by the condition a(n) = max { integer d : alpha^n == 1 (mod d)}. Silverman shows that a(n) is a strong divisibility sequence, that is gcd(a(n), a(m)) = a(gcd(n, m)) for all n and m in N; in particular, if n divides m then a(n) divides a(m). For the present sequence we take alpha = 1 + i. For other examples see A230369, A235450 and (conjecturally) A082630.
%H A230368 J. H. Silverman, <a href="https://www.math.uni-bielefeld.de/documenta/vol-coates/silverman.pdf">Divisibility sequences and powers of algebraic integers</a>, Documenta Mathematica, Extra Volume: John H. Coates' Sixtieth Birthday (2006) 711-727
%H A230368 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,4,0,0,0,1,0,0,0,-4).
%F A230368 a(4*n) = |(-4)^n - 1| otherwise a(n) = 1.
%F A230368 a(4*n) = 5*A015521(n).
%F A230368 O.g.f.: 1/(1 - 4*x^4) - 1/(1 + x^4) + 1/(1 - x) - 1/(1 - x^4) = x*(-1 -x -x^2 -5*x^3 +3*x^4 +3*x^5 +3*x^6 +5*x^7 +4*x^8 +4*x^9 +4*x^10) / ( (1-x) *(1+x) *(2*x^2+1) *(2*x^2-1) *(x^2+1) *(x^4+1) ).
%F A230368 Recurrence equation: a(n) = 4*a(n-4) + a(n-8) - 4*a(n-12).
%p A230368 seq( gcd( 1/2*((1 - I)^n + (1 + I)^n - 2), I/2*((1 + I)^n - (1 - I )^n ) ), n = 1..80);
%Y A230368 Cf. A247281, A014985, A015521, A082630, A230369, A235450.
%K A230368 nonn,easy
%O A230368 1,4
%A A230368 _Peter Bala_, Jan 10 2014