Re: An interesting pyramid sequence.
From: Dan (30pack_at_sbcglobal.net)
Date: 06/27/04
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Date: 27 Jun 2004 04:17:42 -0700
30pack@sbcglobal.net (Dan) wrote in message news:<fefe1afc.0406262213.1d5c79cd@posting.google.com>...
> I can print this out in both formats, as a continuing
> sequence or starting with left diagonal it places all
> diagonals in column form.
> I set it up this way here for an easy explanation on how
> numbers are generated.
>
> 1
> 2 3
> 4 5 6
> 7 9 11 12
> 13 16 20 23 24
> 25 29 36 43 47 48
> 49 54 65 79 90 95 96
> 97 103 119 144 169 185 191 192
> 193 200 222 263 313 354 376 383 384
> 385 393 422 485 576 667 730 759 767 768
> 769 778 815 907 1061 1243 1397 1489 1526 1535 1536
> ... etc.
> Where starting with the 3rd row all the inner triangles are
> created by summing the two diagonal integers above it and
> placing this sum beneath and in the center of the two integers
> above.
> With the help of two arrays the algorithm works this way--
> If there is not two integers diagonally above the spot,
> it just terminates the row by incrementing by (1) from the previous
> sum and starts a new row by incrementing by (1) more.
>
> Where the Pascal triangle has a ratio of 2 between the sum
> of each row, the ratio in this sequence converges (2)<---(5).
> The last # in each row divided by the first # of the same row
> converges (1.5)---->(2). So you have a convergence on (2) working
> both ways!
> Also the right diagonal starting with (3) doubles. Each of
> their squares falls on their downward left diagonal that ends in a
> column left of center.
> Other than the right diagonal, I believe the highest integer that
> has a square later in the sequence is (13). This is hard to prove!
>
> I am not sure but believe there is only (5) triangle numbers,
> [1,2,3,6,36], in this sequence--->oo. Also hard to prove!
>
> Into row 54 it goes into scientific notation because each integer
> becomes >16 digits in length.
>
> There's other interesting stuff but I will end it here.
>
> Sequence not in OEIS.
>
> Dan
A correction above-- there is only (4) triangle numbers -- [1,3,6,36]
in this sequence.
Dan
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