What do the "coefficients" in R's HoltWinters function represent? I'm using the HoltWinters function in R and I'm trying to understand what the "coefficients" represent in the object that is returned by that function.  They don't seem to match in any obvious way the values returned when you look at the $
 A: +1 this is confusing. If your time series has length $N$ and frequency $p$, then the so-called "coefficients" (which can be accessed as HW$coeff if HW is the object returned by HoltWinters) are exactly the values $a[N]$, $b[N]$ and $s[N-p+1]$, $s[N-p+2], \cdots s[N]$ where these are defined by the formulas in the Holt Winters help page, which can be accessed from within R with ?HoltWinters.
For the additive model, which is the default, suppose my.ts is a time series object with positive frequency $p$. The values of $a[N-1]$, $b[N-1]$ and all the earlier $s[t]$ up to $s[N-p]$ are given in the table HoltWinters(my.ts)$fitted. The values in HoltWinters(my.ts)$coeff are calculated from these using the formulas
$$a[t] = α (Y[t] - s[t-p]) + (1-α) (a[t-1] + b[t-1])$$
$$b[t] = β (a[t] - a[t-1]) + (1-β) b[t-1]$$
with $t=N$ and $\alpha = $ HoltWinters(my.ts)$alpha, $\beta = $ HoltWinters(my.ts)$beta, and 
$$s[t] = γ (Y[t] - a[t]) + (1-γ) s[t-p]$$
with $t=N-p+1, \ldots, N$ and $\alpha = $ HoltWinters(my.ts)$alpha, $\beta = $ HoltWinters(my.ts)$beta, and $\gamma = $ HoltWinters(my.ts)$gamma.
This works for $a$ and $b$ (the level and trend) but when I do the calculation for the seasonals, I get slightly different values (within 5% or so) than are given in the output. I hope somebody can edit this answer to explain what is going on with the seasonals. Here is a link to the C code for the hw function which is called by the HoltWinters function:
https://svn.r-project.org/R/trunk/src/library/stats/src/HoltWinters.c
A: I agree that there is puzzle. To see the puzzle I considered 
the co2 series available in R. The answer is long. May you just to the *-part that I added today
I have expected that
co2HWBis$coefficients[2]

equals 
co2HW$fitted[length(co2HW$fitted[,3]),3]

i.e. the coeff equals the last extimated trend.
Below you can check that this is not the case. 
However 
co2HW$fitted[length(co2HW$fitted[,3]),3]

equals coefficient you were to obtain if you drop the last value 
of the series as is deomestrated below. I suspect that the coefficient is 
somehow "written forward". I find it furthermore puzzling the matters are 
different if you allow beta to be estimated.
I reading the source code 
(http://svn.r-project.org/R/trunk/src/library/stats/R/HoltWinters.R)
but I am not yet sure what goes on.
This is the complete code
    rm(list=ls()) 

    co2HW = HoltWinters(co2, alpha = 0.2, gamma = 0.2, beta = 0.5)
    # co2HW$coeff[2]
        co2HW$fitted[length(co2HW$fitted[,3]),3]

    co2Bis = window(co2,end=c(1997,11))
    co2HWBis = HoltWinters(co2Bis, alpha=0.2, gamma=0.2, beta=0.5)
    co2HWBis$coefficients[2] 
        # co2HWBis$fitted[length(co2HWBis$fitted[,3])-1,3]

    co2HW$beta*(co2HW$fitted[length(co2HW$fitted[,2]),2] -  
        co2HW$fitted[length(co2HW$fitted[,2])-1,2]) +
        (1 - co2HW$beta)*co2HW$fitted[length(co2HW$fitted[,3])-1,3]

    #####################



    co2HW = HoltWinters(co2, alpha = 0.2, gamma = 0.2)
    # co2HW$coeff[2]
        co2HW$fitted[length(co2HW$fitted[,3]),3]

    co2Bis = window(co2,end=c(1997,11))
    co2HWBis = HoltWinters(co2Bis, alpha=0.2, gamma=0.2)
    co2HWBis$coefficients[2] 
        # co2HWBis$fitted[length(co2HWBis$fitted[,3])-1,3]

    co2HW$beta*(co2HW$fitted[length(co2HW$fitted[,2]),2] -  
        co2HW$fitted[length(co2HW$fitted[,2])-1,2]) +
        (1 - co2HW$beta)*co2HW$fitted[length(co2HW$fitted[,3])-1,3]

*-part
... one night later I think I can give an answer that looks like an answer. In my opinion the problem is the timing of the table co2HW$fitted. The last line is not the estimated trend level and saison of the last period in the sample. The coefficients are the estimated level, trend and saison of the last period but these value are not displayed in the table. I hope th following code is convincing 
rm(list=ls()) 
x = co2
m = HoltWinters(x)
# m$fitted[length(m$fitted[,3]),3]

aux1 = m$alpha*( x[length(x)] - m$fitted[length(m$fitted[,3]),4]  ) + 
    ( 1 - m$alpha )*( m$fitted[length(m$fitted[,3]),3] + 
m$fitted[length(m$fitted[,3]),2] );
aux1
m$coeff[1]

aux2 = m$beta*(aux1 - m$fitted[length(m$fitted[,3]),2] ) +
     (1-m$beta)*m$fitted[length(m$fitted[,3]),3]
aux2
m$coeff[2]

m$coeff[14]
    aux3 = m$gamma*(x[length(x)] - aux1) + 
( 1 - m$gamma )*m$fitted[length(m$fitted[,3]),4]
aux3

A: I think the key point about the coefficients, which I couldn't see in the other answers but may have missed, is that they are the values of smoothed level and  smoothed trend for the last period in the time series on which the forecast was based/made;  and smoothed seasonal components for the last 12 months of that time series. 
Understanding the table of fitted values for the forecast also helps. For each row corresponding to time t, the values of level and trend are the smoothed values for time t-1, and the value of season is the smoothed value for t-p. These are added to give the estimated true level for time t, Xhat.
I have only begun to use R fairly recently, so apologies if my terminology isn't fully accurate. 
A: This is from the HoltWinters documentation in R. I had the same question and this answers why I could not calculate the same seasonal values. The function is using a decomposition method to find all the initial values when incorporating seasonality, whereas for single and double exponential smoothing it doesn't do this.
"For seasonal models, start values for a, b and s are inferred by performing a simple decomposition in trend and seasonal component using moving averages (see function decompose) on the start.periods first periods (a simple linear regression on the trend component is used for starting level and trend). For level/trend-models (no seasonal component), start values for a and b are x[2] and x[2] - x[1], respectively. For level-only models (ordinary exponential smoothing), the start value for a is x[1]."
Found this website that explains how to get initial values: https://robjhyndman.com/hyndsight/hw-initialization/
A: Coefficients in Holtwinter's  Method
In the HoltWinter's model with seasonal= 'additive', I am able to produce the model output for coefficients manually. It seems like we need to use a[N-p],b[N-p] instead of a[N-1],b[N-1] in the HW formula. This may be because 'a' and 'b' depend on 's' component. I think the coefficients are the smoothed out values for the Nth data and for the last season. If the season has length p, then there will be p many smoothed values.
result=HoltWinters(co2,alpha=NULL,beta=NULL,gamma=NULL,seasonal = 'additive')
result$coef
let us produce the coefficients manually
alphat=0.5126484
betat = 0.009497669
gammat=0.4728868
N=length(co2)
p=length(result\$coef)-2
A_N=alphat*(co2[N]-result\$fitted[N-p,4] )+(1-alphat)*(result\$fitted[N-p,2]+
result\$fitted[N-p,3])
A_N
B_N=betat*(A_N-result\$fitted[N-p,2])+(1-betat)*(result\$fitted[N-p,3])
B_N
S_N=gammat*(co2[N]-A_N)+(1-gammat)*result\$fitted[N-p,4]
S_N
