Truncated triangular number

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A truncated triangular number is an integer n that fulfills the equation f[n] = File:Triangle.gifn - 3File:Triangle.gif3 or simply, f[n] = File:Triangle.gifn - 18

The first positive truncated triangular number occurs for n=6. The truncated triangular numbers under 1000 are:

3, 10, 18, 27, 37, 48, 60, 73, 87, 102, 118, 135, 153, 172, 192, 213, 235, 258, 282, 307, 333, 360, 388, 417, 447, 478, 510, 543, 577, 612, 648, 685, 723, 762, 802, 843, 885, 928, 972

(See sequence A051938 in the Online encyclopedia of integer sequences)

Figurate form of the truncated triangular number

The figurate form of these numbers is a triangle of size n, with the 3 corner triangles of size 6 each truncated (cut off).

Truncated triangles and chemistry

37 is the truncated triangular form of File:Triangle.gif10 = 55, since 37 = 55 - 18. Another relationship between 37 and 55 appears in the opening words of Ecclesiastes "All is vaporous"<ref name="eccles1">Ecclesiastes 1:2</ref> (הַכֹּל הָבֶל), alluding to the "vaporous" nature of the elements, as follows.

The 3 numbers preceding 37 in the series are 10, 18, and 27, whose sum is 55. The value of the 4 consecutive numbers in the series 10, 18, 27, and 37 is 92, the number of naturally occuring elements (from hydrogen to uranium).

References

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