SUMS OF SQUARES AND SUMS OF TRIANGULAR NUMBERS INDUCED BY PARTITIONS OF 8 NAYANDEEP DEKA BARUAH, SHAUN COOPER AND MICHAEL HIRSCHHORN Abstract. Let r k (n) and t k (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 t8(n)= 1 2 10 × 3 2 (r8(8n + 8) - 16r8(2n + 2)) , 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. Addresses: Nayandeep Deka Baruah Department of Mathematical Sciences, Tezpur University Napaam-784028, Sonitpur, Assam, India nayan@tezu.ernet.in Shaun Cooper Institute of Information and Mathematical Sciences, Massey University Private Bag 102904, North Shore Mail Centre, Auckland, New Zealand s.cooper@massey.ac.nz Michael Hirschhorn School of Mathematics and Statistics, University of New South Wales Sydney 2052, Australia m.hirschhorn@unsw.edu.au 1. Introduction Let r k (n) and t k (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. 2000 Mathematics Subject Classification. Primary—11E25; Secondary—05A19, 11D85, 33D15. Key words and phrases. sum of squares, sum of triangular numbers, partitions. Nayandeep Deka Baruah is partially supported by BOYSCAST Fellowship grant SR/BY/M-03/05 from DST, Govt. of India. Status: Published in International Journal of Number Theory 4 (2008), no. 4, 525–538. 1