Number Pattern. Determine the pattern and write the next 3 rows?

Shift right and add. Shift right and add. Sorry had to stick those + signs in front otherwise Yahoo doesn't format it correctly.

1 1 1

+ 1 1 1

+ + 1 1 1

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1 2 3 2 1

1 2 3 2 1

+ 1 2 3 2 1

+ + 1 2 3 2 1

-----------------------

1 3 6 7 6 3 1

See the pattern? Therefore the next line is

1 4 10 16 19 16 10 4 1

+ 1 4 10 16 19 16 10 4 1

+ + 1 4 10 16 19 16 10 4 1

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1 5 15 30 45 51 45 30 15 5 1

You can the figure out the next 2 lines now.

1 5 15 30 45 51 45 30 15 5 1

1 6 21 50 90 126 141 126 90 50 21 6 1

1 7 27 77 161 266 367 393 367 266 161 77 27 1

1 5 15 30 45 51 45 30 15 1

This sequence is formed by adding up 3 adjacent values of the previous sequence. Each value in the sequence is the sum of the three values in the previous sequence occupying the same position, the previous position, and the position before that. Zero is used for non-existant values of the previous sequence. Mathematically:

S[k+1](n) = S[k](n) + S[k](n-1) + S[k](n-2)

where S[k] is the kth sequence and S[k](n) is the nth value of the kth sequence. Using the intial sequence of:

S1(1) = 1, S1(2) = 1, S1(3) = 1

you can see the next sequence is:

S2(1) = 1+0+0 = 1

S2(2) = 1+1+0 = 2

S2(3) = 1+1+1 = 3

S2(4) = 0+1+1 = 2

S2(5) = 0+0+1 = 1

For the next sequence after the last one givien in the problem:

S5(1) = S4(1) = 1

S5(2) = S4(2) + S4(1) = 4+1 = 5

S5(3) = S4(3) + S4(2) + S4(1) = 10+4+1 = 15

S5(4) = S4(4) + S4(3) + S4(2) = 16+10+4 = 30

S5(5) = S4(5) + S4(4) + S4(3) = 19+16+10 = 45

S5(6) = S4(6) + S4(5) + S4(4) = 16+19+16 = 51

and it starts declining after that

I think this is a psychic question, not math. I'm not claitrvoyant.

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