This site is supported by donations to The OEIS Foundation.

# Arithmetic progressions

$N\,$ -terms arithmetic progressions are sequences of the form (with $N\,\to \,\infty \,$ for an infinity of terms)

$a(n)=mn+c,\quad n=\{0,\ldots ,N-1\},\,$ where $m\,$ and $c\,$ are constants; therefore $a(n)-a(n-1)\,=\,m\,$ and $a(n)\,\equiv \,c{\pmod {m}}\,$ . For example, {4, 16, 28, 40, 52, 64, 76, 88, 100, 112, ...} (A017569) is an arithmetic progression with $m\,=\,12\,$ and $c\,=\,4\,$ . In terms of growth of sequences, nonconstant arithmetic progressions have linear growth.

Equivalently, a sequence is an arithmetic progression when each term is the arithmetic mean of the neighboring terms, i.e.

$a(n)={\frac {a(n-1)+a(n+1)}{2}}.\,$ ## "Primitive" versus "nonprimitive" arithmetic progressions

An arithmetic progression might be said to be "primitive" if $m\,$ and $c\,$ are coprime. An arithmetic progression with $\gcd(m,\,c)\,=\,k,\,k\,>\,1\,$ (cf. gcd), which might thus be said to be "nonprimitive", is $k\,$ times the corresponding "primitive" arithmetic progression. For example {4, 16, 28, 40, 52, 64, 76, 88, 100, 112, ...} = 4 × {1, 4, 7, 10, 13, 16, 19, 22, 25, 28, ...}.

## Recurrence

$a(n+1)=a(n)+m\,$ ## Generating functions

Arithmetic progressions have rational [ordinary] generating functions of the form

$G_{\{mn+c\}_{n=1}^{\infty }}(x)\equiv \sum _{n=1}^{\infty }(mn+c)\,x^{n}=-\,{\frac {c\,x^{2}-(m+c)\,x}{(x-1)^{2}}}.\,$ 