I Create an Equation for 3 Offspring Rabbits

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I know that the Fibonnaci equation, Fn = Fn-1 + Fn-2, allows you to calculate (If you are using rabbits) each rabbit giving offspring to one other pair or rabbits. How does one create an equation for three offspring for each rabbit, instead of one? Instead of 0, 1, 1, 2, 3, 5, 8, 13..you have 0, 1, 1, 4, 7, 19, 38. I've been trying to create a formula that matches this, but I'm not sure if I'm doing more work than I need to. If k represents the number of offspring, the closest I have come up with is Xn = X(n-1) + X(n-2) + k + n with n being the generation number, Xn being the number of rabbits. It works for the first few generations but falls apart eventually.
 
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I am not sure I understand your example with the rabbits.
It looks like you are saying instead of ##X_n = X_{n-1} + X_{n-2}## you are changing the game to ##X_n = X_{n-1} +3 X_{n-2}.## But I am not sure how you generated your example sequence...since that method doesn't give 38.
 
Oh lawdy. You're right! That 6th generation should be 40. Not 38. And yes, that equation was exactly what I was aiming for. Much appreciated!
 

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