PASCAL’S TRIANGLE FROM POWERS OF TWO

PASCAL’S TRIANGLE FROM POWERS OF TWO

The Pascal’s Triangle can be derived from the powers of two.  This is how.

The leftmost column in the chart below shows, obviously, the powers of 2 in ascending order.

Each number in the second column is the sum of the number above and the number to the left.

Likewise each successive column.

Take the diagonal that rises upwards to the right from each power of two.

The differences between the items in a diagonal equate to a row of the Pascal’s Triangle.  Examples follow the chart.

0                              0              0              0              0              0

1                              1              1              1              1              1

2                              3              4              5              6              7

4                              7              11           16           22           29

8                              15           26           42           64           93

16                           31           57           99           163         256

32                           63           120         219         382         638

64                           127         247         466         848         1486

128                         255         502         968         1816       3302

256                         511         1013       1981       3797       7099

First Example:

The items in a diagonal that starts with 128 (2^7) include:  128, 127, 120, 99, 64, and 29.

The differences between these items equate to row 7 in the Pascal’s Triangle:  1, 7, 21, 35, 35 . . . .

Second Example:

The items in a diagonal that starts with 256 (2^8) include:  256, 255, 247, 219, 163, and 93.

The differences between these items equate to row 8 in the Pascal’s Triangle:  1, 8, 28, 56, 70 . . .

Just a reminder, the rows of the Pascal’s Triangle sum to increasing powers of 2.  And now you know the reverse can happen:  the powers of two can produce the Pascal’s Triangle.

Since the diagonals in the Pascal’s Triangle sum to the Fibonacci numbers, which approach the golden ratio, I guess it is acceptable to say that you can find the golden ratio in powers of two, by this circuitous route of finding the Pascal’s Triangle.

Posted on May 29, 2015

 

 

 

 

 

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