In `stats::arima`, the first residual of an AR(1) model is obtained as a byproduct of the Kalman filter. Example for an AR(1) model:

    # generate 120 observations from an AR(1) model
    set.seed(123)
    y <- arima.sim(n = 120, model = list(ar=0.6))
    # fit the model
    fit <- arima(y, order = c(1,0,0), include.mean = FALSE)
    # get the state space representaton of the fitted model and 
    # run the Kalman filter
    ss <- makeARIMA(phi = coef(fit), theta = numeric(0), Delta = numeric(0))
    kf <- KalmanRun(y = y, mod = ss)
    # residuals
    head(as.vector(residuals(fit)))
    # [1]  0.5017314 -0.5510861  1.7855220  0.5106597 -1.9551303  0.6932320
    head(kf$resid)
# [1]  0.5017314 -0.5510861  1.7855220  0.5106597 -1.9551303  0.6932320
all.equal(as.vector(residuals(fit)), kf$resid)
    # [1] TRUE

For some introduction on how the Kalman filter operates on ARMA models you may see, for example, [this post](http://stats.stackexchange.com/questions/202903/start-up-values-for-the-kalman-filter/207805#207805).