Daniel Dockery

animî nostrî dêbent interdum âlûcinâri

Home of published musician, recording artist, mathematician, programmer, translator, artist, classicist, and general polymath.

Polygorials

June 21st, 2003

A little while ago I ran across and couldn’t resist the urge to acquire a copy of the last published critical edition of Nicomachus of Gerasa‘s nearly 2000-year-old Introduction to Arithmetic (Αριθμητικη εισαγωγη), the Hoche edition published by Teubner in Leipzig in 1866.

Cover of 1866 Introduction to Arithmetic

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.

For those interested, a small, introductory paper on the Polygorials details the work.


Integer sequences

September 15th, 2001

On a mailing list, my attention was caught by a comment,

Daniel Dockery, please help me find better sequences.

Unable, or at least unwilling, to resist, I offered the following integer sequence:


                                                                      7
                                                                     47
                                                                    247
                                                                   1247
                                                                   6247
                                                                  31247
                                                                 156247
                                                                 781247
                                                                3906247
                                                               19531247
                                                               97656247
                                                              488281247
                                                             2441406247
                                                            12207031247
                                                            61035156247
                                                           305175781247
                                                          1525878906247
                                                          7629394531247
                                                         38146972656247
                                                        190734863281247
                                                        953674316406247
                                                       4768371582031247
                                                      23841857910156247
                                                     119209289550781247
                                                     596046447753906247
                                                    2980232238769531247
                                                   14901161193847656247
                                                   74505805969238281247
                                                  372529029846191406247
                                                 1862645149230957031247
                                                 9313225746154785156247
                                                46566128730773925781247
                                               232830643653869628906247
                                              1164153218269348144531247
                                              5820766091346740722656247
                                             29103830456733703613281247
                                            145519152283668518066406247
                                            727595761418342590332031247
                                           3637978807091712951660156247
                                          18189894035458564758300781247
                                          90949470177292823791503906247
                                         454747350886464118957519531247
                                        2273736754432320594787597656247
                                       11368683772161602973937988281247
                                       56843418860808014869689941406247
                                      284217094304040074348449707031247
                                     1421085471520200371742248535156247
                                     7105427357601001858711242675781247
                                    35527136788005009293556213378906247
                                   177635683940025046467781066894531247
                                   888178419700125232338905334472656247
                                  4440892098500626161694526672363281247
                                 22204460492503130808472633361816406247
                                111022302462515654042363166809082031247
                                555111512312578270211815834045410156247
                               2775557561562891351059079170227050781247
                              13877787807814456755295395851135253906247
                              69388939039072283776476979255676269531247
                             346944695195361418882384896278381347656247
                            1734723475976807094411924481391906738281247
                            8673617379884035472059622406959533691406247
                           43368086899420177360298112034797668457031247
                          216840434497100886801490560173988342285156247
                         1084202172485504434007452800869941711425781247
                         5421010862427522170037264004349708557128906247
                        27105054312137610850186320021748542785644531247
                       135525271560688054250931600108742713928222656247
                       677626357803440271254658000543713569641113281247
                      3388131789017201356273290002718567848205566406247
                     16940658945086006781366450013592839241027832031247
                     84703294725430033906832250067964196205139160156247
                    423516473627150169534161250339820981025695800781247
                   2117582368135750847670806251699104905128479003906247
                  10587911840678754238354031258495524525642395019531247
                  52939559203393771191770156292477622628211975097656247
                 264697796016968855958850781462388113141059875488281247
                1323488980084844279794253907311940565705299377441406247
                6617444900424221398971269536559702828526496887207031247
               33087224502121106994856347682798514142632484436035156247
              165436122510605534974281738413992570713162422180175781247
              827180612553027674871408692069962853565812110900878906247
             4135903062765138374357043460349814267829060554504394531247
            20679515313825691871785217301749071339145302772521972656247
           103397576569128459358926086508745356695726513862609863281247
           516987882845642296794630432543726783478632569313049316406247
          2584939414228211483973152162718633917393162846565246582031247
         12924697071141057419865760813593169586965814232826232910156247
         64623485355705287099328804067965847934829071164131164550781247
        323117426778526435496644020339829239674145355820655822753906247
       1615587133892632177483220101699146198370726779103279113769531247
       8077935669463160887416100508495730991853633895516395568847656247
      40389678347315804437080502542478654959268169477581977844238281247
     201948391736579022185402512712393274796340847387909889221191406247
    1009741958682895110927012563561966373981704236939549446105957031247
    5048709793414475554635062817809831869908521184697747230529785156247
   25243548967072377773175314089049159349542605923488736152648925781247
  126217744835361888865876570445245796747713029617443680763244628906247
  631088724176809444329382852226228983738565148087218403816223144531247
 3155443620884047221646914261131144918692825740436092019081115722656247
15777218104420236108234571305655724593464128702180460095405578613281247

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).

An integer sequence
p(5,5x) ÷ 5 for 1 ≤ x ≤ 120

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:


                               1001001
                             101101101101
                          10111111111111101
                        1011112112112112111101
                     101111212212212212212111101
                   10111121222222222222222212111101
                1011112122223223223223223222212111101
              101111212222323323323323323323222212111101
           10111121222232333333333333333333323222212111101
         1011112122223233334334334334334334333323222212111101
      101111212222323333434434434434434434434333323222212111101
    10111121222232333343444444444444444444444434333323222212111101
  1011112122223233334344445445445445445445445444434333323222212111101
101111212222323333434444545545545545545545545545444434333323222212111101
... etc. ...

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.

An integer sequence
f(3,y) × f(2,y-1)

Daniel Dockery

animî nostrî dêbent interdum âlûcinâri

Home of published musician, recording artist, mathematician, programmer, translator, artist, classicist, and general polymath.