Sums of Squares and Triangular Numbers

Let rk(n) and tk(n) denote the number of representations of n as a sum of k squares, and as a sum of k triangular numbers, respectively. We give a generalization of the result rk(8n + k) = cktk(n), which holds for 1 ≤ k ≤ 7, where ck is a constant that depends only on k. Two proofs are provided. One involves generating functions and the other is combinatorial.

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Mixed sums of squares and triangular numbers

Acta Arithmetica ◽

10.4064/aa127-2-1 ◽

2007 ◽

Vol 127 (2) ◽

pp. 103-113 ◽

Cited By ~ 12

Author(s):

Zhi-Wei Sun

Keyword(s):

Sums Of Squares ◽

Triangular Numbers

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On almost universal mixed sums of squares and triangular numbers

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-2010-05290-0 ◽

2010 ◽

Vol 362 (12) ◽

pp. 6425-6425 ◽

Cited By ~ 7

Author(s):

Ben Kane ◽

Zhi-Wei Sun

Keyword(s):

Sums Of Squares ◽

Triangular Numbers

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On Sums of Triangular Numbers and Sums of Squares

American Mathematical Monthly ◽

10.2307/2324243 ◽

1992 ◽

Vol 99 (8) ◽

pp. 752 ◽

Cited By ~ 1

Author(s):

John A. Ewell

Keyword(s):

Sums Of Squares ◽

Triangular Numbers

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SUMS OF SQUARES AND SUMS OF TRIANGULAR NUMBERS INDUCED BY PARTITIONS OF 8

International Journal of Number Theory ◽

10.1142/s179304210800150x ◽

2008 ◽

Vol 04 (04) ◽

pp. 525-538 ◽

Cited By ~ 13

Author(s):

NAYANDEEP DEKA BARUAH ◽

SHAUN COOPER ◽

MICHAEL HIRSCHHORN

Keyword(s):

Sums Of Squares ◽

Triangular Numbers

Let rk(n) and tk(n) denote the number of representations of an integer n as a sum of k squares, and as a sum of k triangular numbers, respectively. We prove that [Formula: see text] and therefore the study of the sequence t8(n) is reduced to the study of subsequences of r8(n). We give an additional 21 analogous results for sums of squares and sums of triangular numbers induced by partitions of 8. We give a brief indication of what happens for the case k ≥ 9.

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