Study Objectives: We investigate a Mamba-based deep learning approach for sleep staging on signals from ANNE One (Sibel Health, Evanston, IL), a non-intrusive dual-module wireless wearable system measuring chest electrocardiography (ECG), triaxial accelerometry, and chest temperature, and finger photoplethysmography and finger temperature. Methods: We obtained wearable sensor recordings from 357 adults undergoing concurrent polysomnography (PSG) at a tertiary care sleep lab. Each PSG recording was manually scored and these annotations served as ground truth labels for training and evaluation of our models. PSG and wearable sensor data were automatically aligned using their ECG channels with manual confirmation by visual inspection. We trained a Mamba-based recurrent neural network architecture on these recordings. Ensembling of model variants with similar architectures was performed. Results: After ensembling, the model attains a 3-class (wake, non rapid eye movement [NREM] sleep, rapid eye movement [REM] sleep) balanced accuracy of 84.02%, F1 score of 84.23%, Cohen's $κ$ of 72.89%, and a Matthews correlation coefficient (MCC) score of 73.00%; a 4-class (wake, light NREM [N1/N2], deep NREM [N3], REM) balanced accuracy of 75.30%, F1 score of 74.10%, Cohen's $κ$ of 61.51%, and MCC score of 61.95%; a 5-class (wake, N1, N2, N3, REM) balanced accuracy of 65.11%, F1 score of 66.15%, Cohen's $κ$ of 53.23%, MCC score of 54.38%. Conclusions: Our Mamba-based deep learning model can successfully infer major sleep stages from the ANNE One, a wearable system without electroencephalography (EEG), and can be applied to data from adults attending a tertiary care sleep clinic.