Do you produce more sperm after 1 week of abstinence vs 1 year of abstinence
There is an increase by abstinence.


  The within-subject variability of the semen sperm count (n), volume
  (v) and the total number of spermatozoa (N) was studied on 220
  ejaculates from 36 normal subjects after an abstinence of 7 days or
  less.
  
  The three semen characteristics for a given subject were highly
  correlated with the length of abstinence: for an increase in
  abstinence of 1 day there were mean increments of 13 × 106/ml for n,
  0·4 ml for v, and 87 × 106 for N.



1979 - Within-subject variability of human semen in regard to sperm count, volume, total number of spermatozoa and length of abstinence



  Semen analysis was performed on six men after various sexual
  abstinence periods (from 2 to 18 days). The variations in semen
  parameters were analyzed statistically as a function of the duration
  of abstinence.
  
  Lengthy sexual abstinence was found to affect all semen
  characteristics. Semen volume and concentration and total sperm count
  showed significant increases, whereas motility and normal morphology
  decreased significantly with duration of abstinence. Significant
  changes in the percentage of normal sperm forms were observed after
  more than seven days' abstinence.
  
  The study indicates that the influence of long sexual abstinence on
  semen quality varies with the variable considered. With regard to
  fertility, a long abstinence period might induce senescence of
  spermatozoa.



1994 - Effect of long abstinence periods on human sperm quality.


Sperms can die of "old age", as the previous quotes mentioned long abstinence period induces senescence. It is hard to find anything about their lifetime, it is at least 7 days I think.


  For humans, entire process of spermatogenesis takes 74 days. Including
  the transport on ductal system, it takes 3 months. Testes produce 200
  to 300 million spermatozoa daily.



according to wikipedia


So there is a constant rate of sperm production.

I think the following differential equation can describe the change of the sperm count:

$$ \dot{x} = μ - Dx $$

where


x is the sperm count
μ is the production rate regulated by testosteron (about 250 million per day)
D is the death rate which depends on the age of the population (depends on the lifetime, production rate, elapsed time)


So there will be a maximum amount by abstinence. By that amount 

$$ μ = Dx $$

so 

$$ x_{max} = μ /D $$