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 Shapes are too Deep (Posted on 2005-06-01)
What are the next three numbers:

1, 2, 4, 7, 13, 23, 39, 64, 99, 150, 216, __, __, __?

 See The Solution Submitted by Dustin Rating: 3.6000 (5 votes)

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 explanation | Comment 3 of 5 |

As time has already run and I think nobody is working on this sequence, I'm going to give explanation of it.

1,  2,  4,  7,  13,  23,  39,  64,  99,  150,  216

1,   2,  3,  6,   10,  16,  25,  35,  51,   66

1,   13,   4,   6,    9,    101615

Inferior row terms are the difference of consecutive terms in the immediately superior row, while the terms composing this superior row are obtained in the same way.

The inferior is composed by two alternate sequences: odd terms here make a 'triangular' sequence, in which terms increases like the nodes of a sequence of proportional triangles (1, 3, 6, 10, 15...). Even terms are the succession of natural squares (1, 4, 9, 16...).

So, it's done. The next two terms of this deep row will be 25 and 21. In the superior row there will be 91 and 112 and in the sequence 307 and 419.

The title is well chosen, because really shapes show themselves in a deep row, and this shapes are triangles ("too") and squares ("deep"). So "Shapes are too deep".

Nice problem, Dustin.

Edited on June 7, 2005, 3:56 pm
 Posted by armando on 2005-06-07 14:41:51

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