REPRESENTATION NUMBERS BY SUMS OF QUADRATIC FORMS X12 + x1x2 + x22 IN SIXTEEN VARIABLES

Let R(a(1),...,a(8) ; n) denote the number of representations of an integer n by the form a(1)(x(1)(2) + x(1)x(2) + x(2)(2)) + a(2)(x(3)(2) + x(3)x(4) + x(4)(2)) + a(3)(x(5)(2) + x(5)x(6) + x(6)(2)) + a(4)(x(7)(2) + x(7)x(8) + x(8)(2)) + a(5)(x(9)(2) + x(9)x(10) + x(10)(2)) + a(6)(x(11)(2) + x(11)x(12) + x(12)(2)) + a(7)(x(13)(2) + x(13)x(14) + x(14)(2)) + a(8)(x(15)(2) + x(15)x(16) + x(16)(2)) . In this article, we derive formulae for R(1, 2, 2, 2, 2, 2, 2, 2; n), R(1, 1, 1, 2, 2, 2, 2, 2; n), R(1, 1, 1, 1, 1, 2, 2, 2; n) and R(1, 1, 1, 1, 1, 1, 1, 2, n). These formulae are given in terms of the function sigma(7) (n) and the numbers tau(8,2) (n) and tau(8,6) (n) .