A282034 Three-set non-British additive number system read by rows.

1, 2, 3, 4, 50, 51, 52, 53, 54, 5, 10, 15, 20, 25, 100, 125, 200, 225, 300, 325, 400, 425, 500, 525, 600, 625, 700, 725, 800, 825, 900, 925, 1000, 1025, 1100, 1125, 1200, 1225, 1300, 1325, 1400, 1425, 1500, 1525, 1600, 1625, 1700, 1725, 1800, 1825, 1900, 1925

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 3 numbers, one from each row of the following array:

1, 2, 3, 4, 50, 51, 52, 53, 54;

5, 10, 15, 20;

25, 100, 125, 200, 225, 300, 325, 400, 425, 500, 525, 600, 625, 700, 725, 800, 825, 900, 925, 1000, ...

(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. 5.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

A032174 and A282032 are two other examples of additive number systems.

Programs

PARI

Vec(x*(1 + x + 45*x^4 - 45*x^6 - 50*x^9 + 4*x^10 + 54*x^11 + 70*x^14 + 20*x^15) / ((1 - x)^2*(1 + x)) + O(x^50)) \\ Colin Barker, Apr 16 2020

Formula

From Colin Barker, Apr 16 2020: (Start)

G.f.: x*(1 + x + 45*x^4 - 45*x^6 - 50*x^9 + 4*x^10 + 54*x^11 + 70*x^14 + 20*x^15) / ((1 - x)^2*(1 + x)).

a(n) = a(n-1) + a(n-2) - a(n-3) for n>16.

(End)