Sum of series sequence modulo 9


From the book of thoth

On the Book of Thoth  there’s little information on wiki, but there are three big bundles written by Mead (see references). It doesn’t help either that many Tarot information and fictional books are often associated with the name Thoth, probably because it has an occult ring to it and it is not a short nice uncommon name.

The name refers to ancient Egyptian texts associated with Thoth, the Egyptian god of writing and knowledge. (Fowden 1993, p. 57) The book of Thoth has its origin in Egyptian texts, delivered by the ancient Greek.

On wikipedia it says that people think the texts should have been written by Thoth, but that is an invented thought; even the Egyptians knew that a God would not write anything himself.

In mythology Mercury was the Roman name given to Greek Hermes, son of Zeus and brother to the Sun-god Apollo. Hermes has his counterpart in Egypt as Thoth, son of Ra the Sun-god.

The alleged Thoth was scribe to the Sun-god, but also advocate for the dead; this dual role was handed through to Hermes as messenger of the gods and to the souls of the dead.

The relation between the Egyptian Thoth and Hermes is not always clear, but these two names must not be used for the same God, although this happens often.
Thoth is the Egyptian deity, where the later Greek equivalent Hermes Trismegistus is another.  The exact difference is not clear because in Greek literature and the later Latin translation  and commentaries they were often characterized as one and the same.  (see the three volumes by GRS Mead, Ficino and others for more info)

The Antoine Court de Gébelin wiki writes: It was his immediate perception, the first time he saw the Tarot deck, that it held the secrets of the Egyptians.  Without producing any historical evidence, he claimed in 1781 that Egyptian priests had distilled the ancient Book of Thoth into these images of the Tarot deck, which associates the 22 letters of the Hebrew alphabet with  the Tarot’s 21 trumps and the fool. The Trionfi and Visconti (1447) cards are of course well known, but older variants have not been found of possible related cards.

A Tarot deck has 78 cards, based on the sum of series n=12 (1+2+3+4+5+6+7+8+9+10+11+12 = 78).


My primary source was the book review of the book:

Prof. Dr. Woldemar Graf von Uexkuell-Gyllenband – ‘Die Einweihung im alten Aegypten’
Avalun, 1957, Leinen.  Translated: Het Boek Thoth, vertaling door Ds. W.Tj. Klumper.
Uitgegeven door: Geesteswetenschappelijk Centrum Utrecht

on the Dutch site geestkunde. (link)

In this book the author describes the 22 phases to the neophyt (student) during a period of 10 days and 12 nights. After the  XIX the nineteenth statue – the Sun of Osiris, shows the occult numeral values by two methods, which I will call: deduction and merging.

The occult explanation behind the resulting numbers is not very complex but beyond my scope here.

Ich muss aber darauf aufmerksam machen, dass auf dem Weg des Osiris nicht nur die Inhalte der einzelnen Bilder untereinander zusammenhängen, indem sie zugleich alle auf das erste Bild zurückweisen, sondern es stehen auch alle Bildzahlen jedes einzelnen Weges in einem geheimnisvollen, mathematischen Zusammenhang. Alle Bilder des Osirisweges deuten durch ihre Zahlen auf das erste Bild zurück. Es ist jetzt an der Zeit, dich in einen Teil der Geheimnisse unserer heiligen Mathematik einzuführen.

Jede Zahl hat außer ihrem landläufigen Wert noch einen okkulten, der uns ihre tieferen Zusammenhänge offenbart. Um diesen inneren Wert zu finden bedienen wir uns der geheimen Reduktion und Addition. Die geheime Reduktion besteht darin, dass man die einzelnen Ziffern einer Zahl zusammen addiert; das Resultat stellt den okkulten Wert der Zahl dar. Also z. B. der okkulte Wert von 12 ist 3, weil 1+2 = 3 ist.
Die Addition besteht darin, dass man alle Zahlen, die einer Zahl voran gehen, zu ihr hinzuaddiert; die Summe stellt ihren geheimen Wert dar. Z.B.: Der geheime Wert von 12 ist 3, weil 1 + 2 + 3 + 4 + 5 + 6+7 + 8 + 9 + 10 + 11 + 12 = 78 ist; 78 ist aber durch Reduktion 7 + 8 = 15; 15 ist aber wiederum 1 + 5 = 6; 6 ist aber durch geheime Addition 1 + 2 + 3 + 4 + 5 + 6 = 21; 21 ist endlich 2 + 1 = 3. Wir sehen also, dass wir auf beiden Wegen der geheimen Mathematik zu dem gleichen Resultate kommen.


The digits of a number can be added to obtain an occult number. For example 12 => 1+2=3


The numbers that preceded the number will be added to the number itself. For example 12 => 1+2+3+4+5+6+7+8+9+10+11+12 = 78.

We call this method the ‘sum of series’. In the previous example the ‘sum of series n=12’.


Now we can use these methods together according to the method:

If n is the number.

Do for n: calculate sum of series. Then add the digits of the result.
If result<11 then end else n=result and do next.

If n=3 then that is the highest possible number.
If n=10-fold then we reach the end of the sequence and we reset at n=1.
The 9 is a circular number.

Then for n=1+…. the repeating sequence is always :  1,3,3,  1,3,3,  1,9,9


The first numbers in Basic form:

if n=1 then thoth = 1

if 2: thoth = 3

if 3: thoth = 3

if 4: thoth = 1

if 5: thoth = 3

if 6: thoth = 3

if 7: thoth = 1

if 8: thoth = 9

if 9: thoth = 9

if 10: thoth = 1


And here are the results, the red cells show the final number, which were calculated. On the far right the real calculated final Thoth number, as I call them,  are shown.

It it clear that the sequence repeats every 9 numbers.

The sequence has been signed up to the Online Encyclopedia of Integer Sequences as,  but as expected the math people there do understand rolling balls and the math behind it better than something invented 2500 years before Christ.

For example see:

  • example Permutation of the minimal 27 balls of a ball clock after a 12-hour period
  • the y-coordinates of the magic tori that display the Agrippa order-3 “Saturn,” order-5 “Mars,” order-7 “Venus,” order-9 “Luna,” and higher-odd-order-N magic squares, see
  • A007908 Concatenation of the numbers from 1 to n. 146:,  Also called the triangle of the gods…


19-4-2018 -> okay, let’s try to re-formulate the sequence into the mathematical language (-sigh-) and try again to add the sequence to the OEIS.

My “sum of series1” should be

A000217 Triangular numbers or binomial(n+1,2) : a(n) = binomial(n+1,2) = n(n+1)/2 = 0 + 1 + 2 + … + n.

or   A037123 a(n) = a(n-1) + Sum of digits of n.

Then the “sum of digits1”, perhaps a reference can be made to A007953 Digital sum (i.e., sum of digits) of n; also called digsum(n):

Anyways the “sum of digits1” is exact the:  A004157 Sum of digits of n-th triangular number: formula:  a(n) = A007953(A000217(n)).

0, 1, 3, 6, 1, 6, 3, 10, 9, 9, 10, 12, 15, 10, 6, 3, 10, 9, 9, 10, 3, 6, 10, 15, 3, 10, 9, 18, 10, 12, 15, 19, 15, 12, 19, 9, 18, 10, 12, 15, 10, 15, 12, 19, 18, 9, 10, 12, 15, 10, 15, 12, 19, 9, 18, 10, 21, 15, 10, 15, …….

Then the next column “sum of series2” and “sum of digits2” are repeats until the conditions 1….9 are met.

Finally we get the simpler form:

And here the sequence  [1,3,6, 1,6,3, 1,0,0]  with a period of 9  is:  b(n)*b(n+1) mod 9) online here 

sources and links



Relation Thot  & Hermes:

Numbers in Egypt:


Pietro Bongo:  “Nel 1585 pubblicò a Venezia De mystica quaternarii numeri significatione che verrà a far parte dell’opera maggiore del B.: P.Bungi Numerorum mysteria… opus… hac secunda editione ab auctore ipso diligentissime recognitum et tertia parte locupletatum (pubblicata per la prima volta a Bergamo nel 1583 0 1584 – secondo il Graesse – col titolo di De Mystica numerorum significatione). Finalmente, due anni prima di morire, il B. rifece ancora una volta il libro, che uscì con il titolo: Numerorum mystheria,opusmaximarum rerum doctrina et copia repertum,in quo mirus in primis,idemque perpetuus Arithmeticae Pythagoricae cum Divinae Paginae numeris consensus,multiplici ratione probatur (Bergomi 1599). Un’ultima edizione venne fatta a Parigi nel 1618 (cfr. Vaerini).” source

Available on google books.

Thrice-greatest Hermes; studies in Hellenistic theosophy and gnosis, being a translation of the extant sermons and fragments of the Trismegistic literature, with prolegomena, commentaries, and notes, by Hermes, Trismegistus; Mead, G. R. S. (George Robert Stow), 1863-1933.  Publ. date 1906. Internet Archive. Study of the origin of the Christian Faith. link and here

The german version of the book, Prof. Dr. Woldemar Graf von Uexkuell-Gyllenband – ‘Die Einweihung im alten Aegypten’ Avalun, 1957, Leinen:

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