PSR J1813-1749 is one of the most energetic rotation-powered pulsars known, producing a pulsar wind nebula (PWN) and gamma-ray and TeV emission, but whose spin period is only measurable in X-ray. We present analysis of two Chandra datasets that are separated by more than ten years and recent NICER data. The long baseline of the Chandra data allows us to derive a pulsar proper motion mu_R.A.=-(0.067+/-0.010) yr^-1 and mu_decl.=-(0.014+/-0.007) yr^-1 and velocity v_perp~900-1600 km/s (assuming a distance d=3-5 kpc), although we cannot exclude a contribution to the change in measured pulsar position due to a change in brightness structure of the PWN very near the pulsar. We model the PWN and pulsar spectra using an absorbed power law and obtain best-fit absorption NH=(13.1+/-0.9)x10^22 cm^-2, photon index Gamma=1.5+/-0.1, and 0.3-10 keV luminosity Lx~5.4x10^34 erg/s (d/5 kpc)^2 for the PWN and Gamma=1.2+/-0.1 and Lx~9.3x10^33 erg/s (d/5 kpc)^2 for PSR J1813-1749. These values do not change between the 2006 and 2016 observations. We use NICER observations from 2019 to obtain a timing model of PSR J1813-1749, with spin frequency nu=22.35 Hz and spin frequency time derivative nudot=(-6.428+/-0.003)x10^-11 Hz/s. We also fit nu measurements from 2009-2012 and our 2019 value and find a long-term spin-down rate nudot=(-6.3445+/-0.0004)x10^-11 Hz/s. We speculate that the difference in spin-down rates is due to glitch activity or emission mode switching.
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