Field-programmable gate array (FPGA) optimized random number generators (RNGs) are more resource-efficient than software-optimized RNGs because they can take advantage of bitwise operations and FPGA-specific features. However, it is difficult to concisely describe FPGA-optimized RNGs, so they are not commonly used in real-world designs. This paper describes a type of FPGA RNG called a LUT-SR RNG, which takes advantage of bitwise XOR operations and the ability to turn lookup tables (LUTs) into shift registers of varying lengths. This provides a good resource-quality balance compared to previous FPGA-optimized generators, between the previous high-resource high-period LUT-FIFO RNGs and low-resource low-quality LUT-OPT RNGs, with quality comparable to the best software generators. The LUT-SR generators can also be expressed using a simple C++ algorithm contained within this paper, allowing 60 fully-specified LUT-SR RNGs with different characteristics to be embedded in this paper, backed up by an online set of very high speed integrated circuit hardware description language (VHDL) generators and test benches.