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July 3rd, 2018, 02:07 AM   #1
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Numeric sequence

I have a numeric sequence:

l(n)=sum[from k=1 to n](4k-1)*l(n-k)
l(0)=1

Is there a function f(x) that satisfies l(n)?
And if so, what is it?

Last edited by skipjack; July 3rd, 2018 at 10:01 PM.
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July 3rd, 2018, 04:13 PM   #2
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I'm not sure what you mean by "a function f(x) that satisfies l(n).
Since that formula is "well defined", of course there exist a function defined on the positive integers that satisfies that formula.

l(0)= 1 so $\displaystyle l(1)= (4- 1)l(0)= 3$
$\displaystyle l(2)= (4- 1)l(1)+ (8- 1)l(0)= 3(3)+ 7(1)= 16$
$\displaystyle l(3)= (4- 1)l(2)+ (8- 1)l(1)+ (12- 1)l(0)= 4(16)+ 7(3)+ 11(1)= 64+ 21+ 11= 96$
$\displaystyle l(4)= (4- 1)l(3)+ (8- 1)l(2)+ (12- 1)l(1)+ (16- 1)l(0)= 3(96)+ 7(16)+ 11(3)+ 15(1)= 288+ 112+ 33+ 15= 448$.

Continue. No simple formula jumps out at me.

Last edited by skipjack; July 3rd, 2018 at 10:48 PM.
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July 3rd, 2018, 10:48 PM   #3
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You should have had "3(16)" instead of "4(16)", which means that l(3) = 80, l(4) = 3(80) + 7(16) + 11(3) + 15(1) = 400, etc.

That is, l(1) = 3, l(2+m) = 16*5^m for m = 0, 1, 2, etc.
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