Representations of Bell quinary quadratic forms

Let f be a positive definite integral quinary quadratic form and let r(n, f) be the number of representations of an integer n by f. A quinary quadratic form f is said to be a Bell quinary quadratic form if f is isometric to \(x_1^2+2^{\alpha }x_2^2+2^{\beta }x_3^2+2^{\gamma }x_4^2+2^{\delta }x_5^2\) $x_1^2+2^{\alpha }{x}_2^2+2^{\beta }{x}_3^2+2^{\gamma }{x}_4^2+2^{\delta }{x}_5^2$ for some nonnegative integers \(\alpha , \beta , \gamma , \delta \) $\alpha ,\beta ,\gamma ,\delta$ . In this article, we reprove the result of Bell (Am J Math 52:271–279, 1930) and compute r(n, f) by using the number of representations of the sum of 5 squares for any Bell quinary quadratic form f with class number 2.