The provenance of the volume is interesting in itself. A small imprint on the inside front cover, “Paul Koehler—Buchhändler und Antiquar—Leipzig ’05”, shows where, thirty-nine years after its publication, it went through the bookshop of the famed German antiquarian and publisher Paul Koehler; in the rear, on the blank bottom of the last page of the index, is a stamp in Hebrew, stating that it was at one point part of the library collection of the University of Jerusalem; then on the FEP, the handwritten signature of the late, renowned philologist, linguist and classical scholar, Benedict Einarson (long of U. Chicago’s classics dept.). When I found the volume, it had already made its way back to a small bookshop in the UK.
In a recent period of recovery, I kept myself occupied by reading the little volume, and while so engaged found myself entertaining myriad ideas about polygonal numbers, a dormant fascination reawakened by Nicomachus. Skipping a good bit of middle ground and jumping to the point: I generalized the idea of the factorial function, which traditionally produces only the product of the first n natural or counting numbers, to a function which returns the product of the first n polygonal numbers for k-sided polygons. For simplicity of reference, I have called these “polygorials”, merging the two words polygonal factorials. In exploring these I discovered that a few have already appeared in the literature as other sequences in the Encyclopedia of Integer Sequences, though most were novel to this research, which also yielded other surprises, such as the fact that the ratio of the nth hexagorial to the nth trigorial yields the Catalan numbers.
The last digit remains 7; for numbers of more than one digit, the final two digits remain 47; for numbers with more than two digits, the final three remain 247. For numbers of more than three digits, the fourth-to-the-last digit alternates between 1 and 6 (1 for even x [see below, for how to generate the sequence, to see what I mean by x], 6 for odd). For numbers of more than four digits, the fifth-to-the-last digit repeats the sequence: 3, 5, 8, 0. More than five digits, and the sixth-to-last digit repeats: 1, 7, 9, 5, 6, 2, 4, 0. More than six, the seventh-to-last repeats: 3, 9, 7, 8, 1, 7, 5, 5, 8, 4, 2, 3, 6, 2, 0, 0. And so on.
How is it derived? Let p(n,k) designate the nth polygonal number p of the set of polygonal numbers for a k-sided polygon. E.g., p(n,3) would reveal the triangular numbers, p(n,4) the squares, etc. You can define the function in pari with
p(n,k) = (n/2)*(n*k-k+4-2*n)
or in Maple with
p := proc(n,k) (n/2)*(n*k-k+4-2*n) end:
The sequence is generated simply by taking p(5,5x)/5 for x=1,2,3…n (n was stopped arbitrarily at 100 in the above example list).
If you want to skip the polygonal number function altogether, you can render the above sequence with f(x) = 2*5x-3. I included the reference to the polygonal numbers simply to show the source from which I derived the sequence.
In response to the same person having asked about an æsthetically pleasing sequence (to me), I came up with this:
If you define a generalized function f(x,y) → ((10^x)^y - 1) / (10^x - 1), then you can generate the above by evaluating f(3,y) × f(2,y-1) for successive values of y > 2.
In case the “jumble” of numbers aren’t very clear for some, I rendered a quick image of the sequence colored to show the distribution of prime digits: white areas (inside the general triangle shape) represent non-primes, black areas are the digits 0 and 1; primes are in red.