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We perform a numerical study of a new microcanonical polymer model on thethree dimensional cubic lattice, consisting of ideal chains whose range andnumber of nearest-neighbor contacts are fixed to given values. Our simulationssuggest an interesting exact relation concerning the internal energy permonomer of the Interacting Self-Avoiding Walk at the $\theta-$point.

Energy of the Interacting Self-Avoiding Walk at the $\theta-$point