# sample series
x <- AirPassengers
# to illustrate of a more general case,
# take a subsample that does not start in the first season
# and ends in the last season
x <- window(x, start=c(1949,2), end=c(1959,4))

# monthly intercepts
S <- frequency(x)
monthly.means <- do.call("rbind",
  replicate(ceiling(length(x)/S), diag(S), simplify = FALSE))

monthly.means <- ts(monthly.means, frequency = S, start = c(start(x)[1], 1))
monthly.means <- window(monthly.means, start = start(x), end = end(x))

# column names
if (S == 12) {
  colnames(monthly.means) <- month.abb
} else
  colnames(monthly.means) <- paste("season", 1L:S)

monthly.means[1:15,]
#       Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
#  [1,]   0   1   0   0   0   0   0   0   0   0   0   0
#  [2,]   0   0   1   0   0   0   0   0   0   0   0   0
#  [3,]   0   0   0   1   0   0   0   0   0   0   0   0
#  [4,]   0   0   0   0   1   0   0   0   0   0   0   0
#  [5,]   0   0   0   0   0   1   0   0   0   0   0   0
#  [6,]   0   0   0   0   0   0   1   0   0   0   0   0
#  [7,]   0   0   0   0   0   0   0   1   0   0   0   0
#  [8,]   0   0   0   0   0   0   0   0   1   0   0   0
#  [9,]   0   0   0   0   0   0   0   0   0   1   0   0
# [10,]   0   0   0   0   0   0   0   0   0   0   1   0
# [11,]   0   0   0   0   0   0   0   0   0   0   0   1
# [12,]   1   0   0   0   0   0   0   0   0   0   0   0
# [13,]   0   1   0   0   0   0   0   0   0   0   0   0
# [14,]   0   0   1   0   0   0   0   0   0   0   0   0
# [15,]   0   0   0   1   0   0   0   0   0   0   0   0

monthly.trends <- monthly.means * seq_along(x) / S
round(monthly.trends[1:15,], 2)
#       Jan  Feb  Mar  Apr  May  Jun Jul  Aug  Sep  Oct  Nov  Dec
#  [1,]   0 0.08 0.00 0.00 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00
#  [2,]   0 0.00 0.17 0.00 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00
#  [3,]   0 0.00 0.00 0.25 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00
#  [4,]   0 0.00 0.00 0.00 0.33 0.00 0.0 0.00 0.00 0.00 0.00 0.00
#  [5,]   0 0.00 0.00 0.00 0.00 0.42 0.0 0.00 0.00 0.00 0.00 0.00
#  [6,]   0 0.00 0.00 0.00 0.00 0.00 0.5 0.00 0.00 0.00 0.00 0.00
#  [7,]   0 0.00 0.00 0.00 0.00 0.00 0.0 0.58 0.00 0.00 0.00 0.00
#  [8,]   0 0.00 0.00 0.00 0.00 0.00 0.0 0.00 0.67 0.00 0.00 0.00
#  [9,]   0 0.00 0.00 0.00 0.00 0.00 0.0 0.00 0.00 0.75 0.00 0.00
# [10,]   0 0.00 0.00 0.00 0.00 0.00 0.0 0.00 0.00 0.00 0.83 0.00
# [11,]   0 0.00 0.00 0.00 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.92
# [12,]   1 0.00 0.00 0.00 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00
# [13,]   0 1.08 0.00 0.00 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00
# [14,]   0 0.00 1.17 0.00 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00
# [15,]   0 0.00 0.00 1.25 0.00 0.00 0.0 0.00 0.00 0.00 0.00 0.00

set.seed(123)
xreg1 <- runif(length(x), 100, 200)
xreg2 <- rnorm(length(x), mean = mean(x))

summary(lm(x ~ 0 + monthly.means + monthly.trends + xreg1 + xreg2))
# Coefficients:
#                    Estimate Std. Error t value Pr(>|t|)
# monthly.meansJan  439.46067  363.55393   1.209    0.230
# monthly.meansFeb  457.02743  362.72206   1.260    0.211
# monthly.meansMar  470.05809  365.81017   1.285    0.202
# monthly.meansApr  462.42569  364.45517   1.269    0.208
# monthly.meansMay  452.74617  364.39972   1.242    0.217
# monthly.meansJun  453.95303  364.38891   1.246    0.216
# monthly.meansJul  462.79607  365.35861   1.267    0.208
# monthly.meansAug  455.03446  363.19802   1.253    0.213
# monthly.meansSep  453.29026  363.20374   1.248    0.215
# monthly.meansOct  440.57396  363.25728   1.213    0.228
# monthly.meansNov  431.31230  364.77546   1.182    0.240
# monthly.meansDec  443.82401  363.05542   1.222    0.224
# monthly.trendsJan  27.92925    1.52426  18.323   <2e-16 ***
# monthly.trendsFeb  23.58292    1.33391  17.680   <2e-16 ***
# monthly.trendsMar  27.96668    1.35829  20.590   <2e-16 ***
# monthly.trendsApr  27.33010    1.33884  20.413   <2e-16 ***
# monthly.trendsMay  29.04411    1.53939  18.867   <2e-16 ***
# monthly.trendsJun  35.80080    1.52180  23.525   <2e-16 ***
# monthly.trendsJul  39.76516    1.59066  24.999   <2e-16 ***
# monthly.trendsAug  40.53217    1.52845  26.519   <2e-16 ***
# monthly.trendsSep  32.49961    1.54164  21.081   <2e-16 ***
# monthly.trendsOct  28.34616    1.52805  18.551   <2e-16 ***
# monthly.trendsNov  24.20748    1.53510  15.769   <2e-16 ***
# monthly.trendsDec  26.36839    1.53157  17.217   <2e-16 ***
# xreg1               0.05708    0.05222   1.093    0.277
# xreg2              -1.45284    1.45625  -0.998    0.321
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#
# Residual standard error: 13.81 on 97 degrees of freedom
# Multiple R-squared:  0.9979,  Adjusted R-squared:  0.9974
# F-statistic:  1787 on 26 and 97 DF,  p-value: < 2.2e-16

coef.seasonal.intercepts <- coef(lm(x ~ 0 + monthly.means))
sample.means <- lapply(split(x, cycle(x)), mean)
all.equal(coef.seasonal.intercepts,
  as.vector(unlist(sample.means)), check.attributes = FALSE)
# [1] TRUE