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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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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)

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A033077 A190912 A306108 * A111665 A111666 A080475

Adjacent sequences:  A282030 A282031 A282032 * A282034 A282035 A282036

KEYWORD

nonn,tabf,easy

AUTHOR

N. J. A. Sloane, Feb 20 2017

STATUS

approved

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Last modified October 22 07:53 EDT 2021. Contains 348160 sequences. (Running on oeis4.)