A282034 -id:A282034 - cp's OEIS frontend

A282032 Additive number system based on U.S. coins.

1, 2, 3, 4, 5, 10, 15, 20, 25, 50, 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, 3800

Offset: 1

N. J. A. Sloane, Feb 20 2017

Any positive integer can be written uniquely as a sum of at most 5 numbers, one from each row of the following array:
1,2,3,4;
5,10,15,20;
25;
50;
100, 200, 300, 400, 500, ...
(the last row being infinite).

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. 2.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

A032174 and A282034 are two other examples of additive number systems.

A282033 gives a very similar family of sets which is not an additive system.

Vec(x*(1 + 4*x^5 + 20*x^9 + 25*x^10 + 50*x^11) / (1 - x)^2 + O(x^50)) \\ Colin Barker, Apr 16 2020

From Colin Barker, Apr 16 2020: (Start)
G.f.: x*(1 + 4*x^5 + 20*x^9 + 25*x^10 + 50*x^11) / (1 - x)^2.
a(n) = 2*a(n-1) - a(n-2) for n>11.
(End)

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

Original entry on oeis.org

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

N. J. A. Sloane, Feb 20 2017

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).

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).

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

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 *)

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

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)

cp's OEIS Frontend

A282032 Additive number system based on U.S. coins.

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