Residual network (ResNet) and densely connected network (DenseNet) have significantly improved the training efficiency and performance of deep convolutional neural networks (DCNNs) mainly for object classification tasks. In this paper, we propose an efficient network architecture by considering advantages of both networks. The proposed method is integrated into an encoder-decoder DCNN model for medical image segmentation. Our method adds additional skip connections compared to ResNet but uses significantly fewer model parameters than DenseNet. We evaluate the proposed method on a public dataset (ISIC 2018 grand-challenge) for skin lesion segmentation and a local brain MRI dataset. In comparison with ResNet-based, DenseNet-based and attention network (AttnNet) based methods within the same encoder-decoder network structure, our method achieves significantly higher segmentation accuracy with fewer number of model parameters than DenseNet and AttnNet. The code is available on GitHub (GitHub link: https://github.com/MinaJf/DRU-net).