Ponder This Challenge - July 2020 - Sequences generated by letters

Let's play the following game: Start with the letter "Z" and replace it with its British spelling, "Z" -> "ZED". Now take each letter in the resulting string, and again replace it with its spelling ("E" is spelled simply "E" and "D" is spelled "DEE"). We obtain the string "ZEDEDEE". Repeating this once again, we obtain "ZEDEDEEEDEEEE" and so forth. We have obtained the following sequence:
Z → ZED → ZEDEDEE → ZEDEDEEEDEEEE → ...

We now count the sequence of lengths obtained in this manner:
1 → 3 → 7 → 13 → ...

This is the sequence of central polygonal numbers.

This "game" can be played with many different alphabets and spelling choices. For example, in Hebrew we can use the following spelling:
{"A": "ALF", "L": "LMD", "F": "FA", "M": "MM", "D": "DLT", "T": "TAV", "V": "VAV"}
(Here some of the letters are spelled somewhat irregularly.)

Playing the game for "A" gives rise to the sequence 1, 3, 8, 21, 55, 144 which contains every even-placed element of the Fibonacci sequence except the first: a(n) = F(2n) (where n starts from 1 and the Fibonacci sequence is 0, 1, 1, 2, 3,...).

Your goal is to find an "alphabet" (which can be completely invented) which gives rise in this manner to the golden rectangle numbers sequence, a(n)=F(n)*F(n+1) (again, for n starting from 1).

Give your answer as a dictionary of letters and spellings, as given above (as another example, the dictionary for the "ZED" game is {"Z": "ZED", "E": "E", "D": "DEE"}. Use Latin letters even if your solution is based on another alphabet. (Please tell us which one!)

A bonus '*' for finding a subsequence of the Fibonacci numbers in any real-world alphabet (well-accepted invented languages such as Klingon or Sindarin are also accepted).

Solution

• Click here to view the solution

  A possible solution (out of many) is {"A": "BC", "B": "AAB", "C": "ABC"}.

  The simplest example of a real-world alphabet giving rise to sub-sequences of the Fibbonacci sequence is German, where the partial phonetic alphabet {"A":"AH","H":"HAH"} gives rise to the sub-sequences 1, 2, 5, 13,... and 1, 3, 8, 21,...

Solvers

• *Alper Halbutogullari (28/6/2020 04:13 PM IDT)
• Florian Fischer (28/6/2020 04:43 PM IDT)
• *Oscar Volpatti (28/6/2020 07:14 PM IDT)
• *James Dow Allen (28/6/2020 10:14 PM IDT)
• Keith Schneider (29/6/2020 01:11 AM IDT)
• Radu-Alexandru Todor (29/6/2020 02:29 AM IDT)
• Eden Saig (29/6/2020 03:17 AM IDT)
• Anders Kaseorg (29/6/2020 05:02 AM IDT)
• Uoti Urpala (29/6/2020 05:41 AM IDT)
• Guillermo Wildschut (29/6/2020 01:35 PM IDT)
• Willem H Hendriks (29/6/2020 03:38 PM IDT)
• Ben Jones (29/6/2020 05:26 PM IDT)
• Noam Zaks (29/6/2020 05:59 PM IDT)
• Arthur Breitman (30/6/2020 06:36 AM IDT)
• Ricardo Koller (30/6/2020 04:27 PM IDT)
• Todd Will (30/6/2020 05:39 PM IDT)
• Amos Guler (30/6/2020 08:44 PM IDT)
• Dan Dima (30/6/2020 09:01 PM IDT)
• *Bertram Felgenhauer (01/7/2020 03:43 AM IDT)
• Sean Egan (01/7/2020 04:39 AM IDT)
• Harold Gutch (02/7/2020 12:54 AM IDT)
• Ori Weiss (02/7/2020 12:40 PM IDT)
• Graham Hemsley (02/7/2020 03:47 PM IDT)
• Dominik Reichl (02/7/2020 04:44 PM IDT)
• Gary M. Gerken (02/7/2020 09:35 PM IDT)
• David Greer (03/7/2020 01:37 AM IDT)
• Arthur Vause (04/7/2020 07:51 PM IDT)
• JJ Rabeyrin (05/7/2020 02:03 AM IDT)
• Jens Leskinen (05/7/2020 11:40 AM IDT)
• Alex Fleischer (05/7/2020 02:24 PM IDT)
• Lorenz Reichel (05/7/2020 03:24 PM IDT)
• David Friedman (06/7/2020 02:02 AM IDT)
• Loukas Sidiropoulos (06/7/2020 01:35 PM IDT)
• Shouky Dan & Tamir Ganor (07/7/2020 01:18 PM IDT)
• Vladimir Volevich (07/7/2020 06:43 PM IDT)
• Louis Faucon (08/7/2020 09:41 AM IDT)
• Sanandan Swaminathan (08/7/2020 11:00 AM IDT)
• Jacoby Jaeger (09/7/2020 07:34 AM IDT)
• Yongxing Jim Wang (11/7/2020 04:48 AM IDT)
• Daniel Bitin (12/7/2020 04:29 PM IDT)
• Siddhartha Srivastava (13/7/2020 02:47 PM IDT)
• Божидар Стајчић (09/7/2020 01:12 AM IDT)
• Dan Piponi (13/7/2020 10:25 PM IDT)
• *Tyler Mullen (13/7/2020 10:37 PM IDT)
• *Michael Branicky (14/7/2020 04:37 AM IDT)
• Marco Bellocchi (17/7/2020 02:49 AM IDT)
• Kang Jin Cho (18/7/2020 07:48 PM IDT)
• Dieter Beckerle (18/7/2020 11:50 PM IDT)
• Clive Tong (19/7/2020 12:06 PM IDT)
• Evert van Dijken (19/7/2020 10:30 PM IDT)
• Daniel Chong Jyh Tar (20/7/2020 07:43 PM IDT)
• David Smalling (21/7/2020 06:26 AM IDT)
• Michael Schuresko (24/7/2020 01:18 PM IDT)
• Parth Trivedi (24/7/2020 05:39 PM IDT)
• Tom K (25/7/2020 10:44 AM IDT)
• Stanislav Grudnitskiy (25/7/2020 05:11 PM IDT)
• Thomas HUET (26/7/2020 05:26 PM IDT)
• Li Wang (27/7/2020 04:32 PM IDT)
• Zhou Hangbo (27/7/2020 04:35 PM IDT)
• Colin Carroll (27/7/2020 11:48 PM IDT)
• Joerg Schepers (28/7/2020 02:53 AM IDT)
• Dipto Thakurta (29/7/2020 11:35 AM IDT)
• *Boris Alexeev (29/7/2020 07:26 PM IDT)
• Luke Pebody (30/7/2020 11:23 AM IDT)
• Albert Stadler (30/7/2020 02:28 PM IDT)
• Bert Dobbelaere (30/7/2020 03:47 PM IDT)
• Leonid Chindelevitch (31/7/2020 10:41 AM IDT)
• Jun Sakai (31/7/2020 03:35 PM IDT)
• Markus Kottmann (31/7/2020 08:14 PM IDT)
• Li Li (01/8/2020 07:18 AM IDT)
• Amir Sarid (01/8/2020 10:53 PM IDT)
• Motty Porat (01/8/2020 03:52 AM IDT)
• Kai Guttmann (03/8/2020 09:21 PM IDT)
• Reiner Martin (04/8/2020 01:30 AM IDT)