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In high counts, if you somehow allow yourself to get into a situation where you're low on cash, betting only 1/6 of your chips seems like a poor idea to me. While you ought to have enough to split, so as not to misplay your hand, not doubling doesn't make you play the hand "wrong"; it just doesn't get as many chips on the table as you might have hoped for. You do much more damage by betting 1/6 of your remaining funds, when you could be betting much more.
And yes, doubling after splitting IS NOT a very frequent occurrence. We split only 2.8% of all hands (yes, a bit more frequently in higher counts). We double about 10%. Multiply one by the other, and you realize that first splitting and then doubling is clearly not something that you ought to be saving your money for in the situation we're discussing. It would be very foolish.
Don
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