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Number of distinct quadratic residues mod 8^n.
2

%I #30 Mar 27 2024 15:41:44

%S 1,3,12,87,684,5463,43692,349527,2796204,22369623,178956972,

%T 1431655767,11453246124,91625968983,733007751852,5864062014807,

%U 46912496118444,375299968947543,3002399751580332,24019198012642647,192153584101141164

%N Number of distinct quadratic residues mod 8^n.

%C Number of distinct n-digit suffixes of base 8 squares.

%H Vincenzo Librandi, <a href="/A039305/b039305.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,1,-8).

%F a(n) = floor((8^n+10)/6).

%F G.f.: (1-5*x-13*x^2-4*x^3)/((1-x)*(1+x)*(1-8*x)). - _Colin Barker_, Mar 14 2012

%F a(n) = 8*a(n-1) + a(n-2) - 8*a(n-3) for n>0, a(0)=1. - _Vincenzo Librandi_, Apr 22 2012

%t CoefficientList[Series[(1-5*x-13*x^2-4*x^3)/((1-x)*(1+x)*(1-8*x)),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 22 2012 *)

%t Join[{1},LinearRecurrence[{8,1,-8},{3,12,87},30]] (* _Harvey P. Dale_, Feb 10 2015 *)

%o (Magma) I:=[1, 3, 12, 87]; [n le 4 select I[n] else 8*Self(n-1)+Self(n-2)-8*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Apr 22 2012

%Y Cf. A001018.

%K nonn,easy

%O 0,2

%A _David W. Wilson_