### Abstract

We give a variety of results involving s(n), the number of representations of n as a sum of three squares. Using elementary techniques we prove that if 9 + n, s(9^{λ}n) = (3^{λ+1}-1/2-(^{-n}/3)3^{λ+1}-1/2)s(n); via the theory of modular forms of half integer weight we prove the corresponding result with 3 replaced by p, an odd prime. This leads to a formula for s(n) in terms of s(n′), where n′ is the square-free part of n. We also find generating function formulae for various subsequences of {s(n)}, for instance Σs(3n + 2)q^{n} = 12 Π (1 + q^{2n-1})^{2}(1 - q^{6n})^{3}. n≥0 n≥1

Original language | English (US) |
---|---|

Pages (from-to) | 85-101 |

Number of pages | 17 |

Journal | Discrete Mathematics |

Volume | 199 |

Issue number | 1-3 |

State | Published - Mar 28 1999 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

### Cite this

*Discrete Mathematics*,

*199*(1-3), 85-101.

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*Discrete Mathematics*, vol. 199, no. 1-3, pp. 85-101.

**On representations of a number as a sum of three squares.** / Hirschhorn, Michael D.; Sellers, James A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On representations of a number as a sum of three squares

AU - Hirschhorn, Michael D.

AU - Sellers, James A.

PY - 1999/3/28

Y1 - 1999/3/28

N2 - We give a variety of results involving s(n), the number of representations of n as a sum of three squares. Using elementary techniques we prove that if 9 + n, s(9λn) = (3λ+1-1/2-(-n/3)3λ+1-1/2)s(n); via the theory of modular forms of half integer weight we prove the corresponding result with 3 replaced by p, an odd prime. This leads to a formula for s(n) in terms of s(n′), where n′ is the square-free part of n. We also find generating function formulae for various subsequences of {s(n)}, for instance Σs(3n + 2)qn = 12 Π (1 + q2n-1)2(1 - q6n)3. n≥0 n≥1

AB - We give a variety of results involving s(n), the number of representations of n as a sum of three squares. Using elementary techniques we prove that if 9 + n, s(9λn) = (3λ+1-1/2-(-n/3)3λ+1-1/2)s(n); via the theory of modular forms of half integer weight we prove the corresponding result with 3 replaced by p, an odd prime. This leads to a formula for s(n) in terms of s(n′), where n′ is the square-free part of n. We also find generating function formulae for various subsequences of {s(n)}, for instance Σs(3n + 2)qn = 12 Π (1 + q2n-1)2(1 - q6n)3. n≥0 n≥1

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UR - http://www.scopus.com/inward/citedby.url?scp=0043209368&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0043209368

VL - 199

SP - 85

EP - 101

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -