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

1, 2, 3, 4, 5, 10, 20, 25, 50, 75, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000, 2100, 2200, 2300, 2400, 2500, 2600, 2700, 2800, 2900, 3000, 3100, 3200, 3300, 3400, 3500, 3600, 3700

Offset: 1

Author

N. J. A. Sloane, Feb 20 2017

Keywords

• nonn
• tabf
• easy

Comments

The five sets are the following:

1, 2, 3, 4;

5;

10, 20;

25, 50, 75;

100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, ...

(the last set being infinite).

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).

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Michael Maltenfort, Characterizing Additive Systems, The American Mathematical Monthly 124.2 (2017): 132-148. See Fig. 3.
• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A032174, A282032, A282034 are legitimate examples of additive number systems.

Programs

Mathematica

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 *)

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

Formula

From Colin Barker, Apr 16 2020: (Start)

G.f.: x*(1 + 4*x^5 + 5*x^6 - 5*x^7 + 20*x^8 + 75*x^11) / (1 - x)^2.

a(n) = 2*a(n-1) - a(n-2) for n>12.

(End)