Tillson's T3 is a kind of Moving average. Tim Tillson described it in "Technical Analysis of Stocks and Commodities", January 1998. He named his article "Better Moving Averages".
Tillson’s moving average becomes a popular indicator of Technical analysis. Its advantage is that it gets less lag with the price chart and its curve is considerably smoother. By using it, the trader can get early entry and less number of false signals.
T3 moving average formula/calculation looks like follows:
a = 0.7 (but also 0.618); n = number of chosen days, mostly 3 days (If we choose 5 days, it is called T5 MA, if 8 days, it is called T8 MA etc.);
alpha = 2/(n + 1); Ema1 = Ema (Close); Ema2 = Ema (Ema1); Ema3 = Ema (Ema2); Ema4 = Ema (Ema3); Ema5 = Ema (Ema4); Ema6 = Ema (Ema5);
T3 = –(a*a*a) * Ema6 + (3*a*a + 3*a*a*a) * Ema5 + (–6*a*a – 3*a – 3*a*a*a) * Ema4 + (1 + 3*a + a*a*a + 3*a*a) * Ema3
How to use the T3 Moving average:
 We may use it to generate Buy and Sell signals. If the T3 rises, but the price falls until the T3 level, we Buy. If T3 falls but price rises temporary up to T3 level, we Sell.
 It is also possible to use two crossings of Tillson's moving averages  e.g. T3 and T5. If T3 crosses T5 upwards, we Buy. Should it cross downwards, we Sell.
 T3 helps us to determine the prevailing trend, as well. If it rises, there is an Uptrend prevailing and vice versa. We enter the trades just in the direction of the trend. Also T5, T8 etc. can be used to determine the prevailing trend.
Note: According to the source, you can find T3 calculation as GD(GD(GD))), i.e. based on Generalized Dema. In such case, if you choose 0 as the weight in the Generalized Dema calculation, T3 equals to Ema of Ema of Ema. If you choose the number 1, T3 equals to Dema of Dema of Dema.
If you are interested in a deeper study of this technical indicator and prefer ready to serve solutions, this section may be of interest to you. There you can find all the available indicators in Excel file for download.
