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A282033 An example of a collection of five sets (based on U.S. coinage) which is not an additive number system.

%I A282033 #25 Aug 04 2021 15:44:24
%S A282033 1,2,3,4,5,10,20,25,50,75,100,200,300,400,500,600,700,800,900,1000,
%T A282033 1100,1200,1300,1400,1500,1600,1700,1800,1900,2000,2100,2200,2300,
%U A282033 2400,2500,2600,2700,2800,2900,3000,3100,3200,3300,3400,3500,3600,3700
%N A282033 An example of a collection of five sets (based on U.S. coinage) which is not an additive number system.
%C A282033 The five sets are the following:
%C A282033 1, 2, 3, 4;
%C A282033 5;
%C A282033 10, 20;
%C A282033 25, 50, 75;
%C A282033 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, ...
%C A282033 (the last set being infinite).
%C A282033 In contrast to A282032 this is not an additive number system because 26 for example can be represented in two ways as a sum of numbers from distinct sets (26 = 1+5+20 = 1+25).
%H A282033 Colin Barker, <a href="/A282033/b282033.txt">Table of n, a(n) for n = 1..1000</a>
%H A282033 Michael Maltenfort, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.124.2.132">Characterizing Additive Systems</a>, The American Mathematical Monthly 124.2 (2017): 132-148. See Fig. 3.
%H A282033 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F A282033 From _Colin Barker_, Apr 16 2020: (Start)
%F A282033 G.f.: x*(1 + 4*x^5 + 5*x^6 - 5*x^7 + 20*x^8 + 75*x^11) / (1 - x)^2.
%F A282033 a(n) = 2*a(n-1) - a(n-2) for n>12.
%F A282033 (End)
%t A282033 LinearRecurrence[{2,-1},{1,2,3,4,5,10,20,25,50,75,100,200,300,400},50] (* or *) CoefficientList[Series[x (1+4x^5+5x^6-5x^7+ 20x^8+ 75x^11)/ (1-x)^2, {x,0,50}],x] (* _Harvey P. Dale_, Aug 04 2021 *)
%o A282033 (PARI) Vec(x*(1 + 4*x^5 + 5*x^6 - 5*x^7 + 20*x^8 + 75*x^11) / (1 - x)^2 + O(x^50)) \\ _Colin Barker_, Apr 16 2020
%Y A282033 Cf. A032174, A282032, A282034 are legitimate examples of additive number systems.
%K A282033 nonn,tabf,easy
%O A282033 1,2
%A A282033 _N. J. A. Sloane_, Feb 20 2017