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

Author

N. J. A. Sloane, Feb 20 2017

Keywords

• nonn
• tabf
• easy

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).

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.

Programs

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

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)