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

1,2

Comments

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).

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. 2.

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 + 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)

Prog

(PARI) 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

Crossrefs

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.

Sequence in context: A032543 A140730 A273732 * A205962 A134220 A179146

Adjacent sequences: A282029 A282030 A282031 * A282033 A282034 A282035

Keyword

nonn,tabf,easy

Author

N. J. A. Sloane, Feb 20 2017

Status

approved