`ArithmeticProgression`

Describes an arithmetic progession of the form \((d n + s)_{n \in \mathbb{N}}\). Such an arithmetic progressions is defined by a shift \(s\) and the difference of two consecutive terms \(d\).

EXAMPLES:

sage: from rec_sequences.ArithmeticProgression import *
sage: prog = ArithmeticProgression(2, 3)
sage: print(prog)
Arithmetic progression (2*n+3)_n
sage: prog.get_diff(), prog.get_shift()
(2, 3)
sage: 5 in prog
True
sage: 6 in prog
False
sage: prog[2]
7
sage: prog[:5]
[3, 5, 7, 9, 11]

class rec_sequences.ArithmeticProgression.ArithmeticProgression(diff=0, shift=0)

Bases: sage.structure.sage_object.SageObject

Describes an arithmetic progession (diff*n+shift)_n for natural numbers diff, shift.

__contains__(item)

Checks whether the arithmetic progression contains item provided it is a natural number.

INPUT:

item – a natural number

OUTPUT:

True if item is in the progression and False otherwise.

__eq__(prog)

Checks whether two arithmetic progressions are equal.

INPUT:

prog – an arithmetic progression

OUTPUT:

True if the two progressions are equal and False otherwise.

EXAMPLES:

sage: from rec_sequences.ArithmeticProgression import *
sage: prog1 = ArithmeticProgression(2, 3)
sage: prog2 = ArithmeticProgression(2, 3)
sage: prog3 = ArithmeticProgression(2, 1)
sage: prog1 == prog2
True
sage: prog1 == prog3
False

__getitem__(n)

INPUT:

n – a natural number or a slice object

OUTPUT:

Returns the n-th term of the arithmetic progression.

__init__(diff=0, shift=0)

Creates the arithmetic progression \((d n+s)_n\) where \(d\) is the given difference diff and \(s\) is the given shift shift.

INPUT:

diff – the difference of two consecutive values in the progression.
shift – the shift of the arithmetic progression.

OUTPUT:

The described arithmetic progression.

_latex_()

OUTPUT:

A latex representation of the form

\[\{ d n + s : n \in \mathbb{N} \}\]

where \(d\) is the difference of the progression and \(s\) is the shift of the progression.

_repr_()

OUTPUT:

A string representation of the form “Arithmetic progression (d*n+s)_n” where d is the difference of the progression and s is the shift of the progression.

get_diff(): Returns the difference of the arithmetic progression.

get_shift(): Returns the shift of the arithmetic progression.

is_zero()

True if the arithmetic progression is constantly zero.

EXAMPLES:

sage: from rec_sequences.ArithmeticProgression import *
sage: ArithmeticProgression(2, 3).is_zero()
False
sage: ArithmeticProgression(0, 0).is_zero()
True

ArithmeticProgression

`ArithmeticProgression`