SquaresR

The number of ways to write $n$ as a sum of $d$ squares, counting signs and order.

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SquaresR(d, n)$r_d(n)$: ordered, signed $d$-tuples summing to $n$

Typed signatures

Type	Package	Applies
(integer, integer) -> integer	number-theory	always

Domain: Number theory

Details

• By brute-force lattice-point enumeration, so signs and order are counted directly rather than adjusted for after the fact.
• Jacobi's four-square theorem: $r_4(n)=8\sigma_1(n)$ when $4 \nmid n$. See DivisorSigma.

Examples

Basic #

$(\pm1)^2+(\pm2)^2$ in either order

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In$\mathrm{SquaresR}(2, 5)$Out$8$

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In$\mathrm{SquaresR}(2, 25)$Out$12$

No $n\equiv3\pmod4$ is a sum of two squares

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In$\mathrm{SquaresR}(2, 3)$Out$0$

Ways to write 45 as a sum of two squares, with signs and order

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In$\mathrm{SquaresR}(2, 45)$Out$8$

Scope #

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In$\mathrm{SquaresR}(3, 6)$Out$24$

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In$\mathrm{SquaresR}(8, 3)$Out$448$

Applications #

Lattice points in a disc of radius 20

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In$\sum_{k=0}^{400}\mathrm{SquaresR}(2, k)$Out$1\,257$

Properties #

Jacobi: $r_4(n)=8\sigma_1(n)$ for $4\nmid n$

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In$\mathrm{SquaresR}(4, 10)$Out$144$

$100=0^2+10^2=6^2+8^2$, counting signs and order

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In$\mathrm{SquaresR}(2, 100)$Out$12$

About

Examples	9
Checked against	Wolfram Language 9 agree
Crosswalk	FungrimWolfram Language

See also: PowersRepresentations, DivisorSigma