Belgian Words

Belgian Words are words which follow, in a certain way, the rule of the Belgian Numbers (formerly Eric Numbers).

What is a “Belgian Number”? [reminder]

176 is a “Belgian number” because, starting from 0, one can build a sequence containing 176 in this way:

0 1 8 14 15 22 28 29 36 42 43 50 ... 155 162 168 169 176. (building sequence)

1 7 6  1  7  6  1  7  6  1  7  ...     7   6   1   7   (first differences repeated pattern)

The “first differences” building rule is easy to understand. The above example shows that you don’t have to add the full digit-pattern [1+7+6] to produce the according Belgian number: 176 appears when 7 is added to the previous integer – not 6.

Here are the first Belgian numbers:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 17 18 20 21 22 24 26 27 30 31 33 35 36 39 40 42 44 45 48 50 53 54 55 60 62 63 66 70 71 72 77 80 81 84 88 90 93 99 100 101 102 106 108 110 111 112 114 117 120 ...

Now, let us transform a word into a number by substituting each letter with it’s rank in the alphabet; the word ERIC, for instance, becomes 51893:

E=5, R=18, I=9, C=3 --> ERIC = 51893

ERIC will be an Belgian Word if and only if the number 51893 is part of the sequence which starts with 0 and builds up step by step with the pattern E+R+I+C (5+18+9+3) been added:

0 5 23 32 35 40 58 67 70 75 93 102... (building sequence)

E R  I  C  E  R  I  C  E  R  I... (first differences repeated pattern)

In the end we have:

... 51858 51867 51870 51875 51893 <-- hit! ERIC is a Belgian Word.

... I     C     E     R

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What about Self-Belgian Word Numbers (SBWN)? Self-Belgian Word Numbers would be Belgian Words which describe their own seed...

CINQ (five) is such a word in French (we have not yet searched for English ones):

First, we transform it into a number:

C=3, I=9, N=14, Q=17  --> CINQ = 39147

Then we start the building sequence with the according “seed” 5:

5 8 17 31 48 51 60 74 91 94 103 ... (building sequence)

C I  N  Q  C  I  N  Q  C  I ... (first differences repeated pattern)

In the end we have:

...39118 39135 39138 39147 <-- hit! CINQ is a French SBWN.

Q     C     I

VINGT (twenty, in French) fits also:

V=22, I=9, N=14, G=7, T=20  --> VINGT = 22914720

Building sequence with seed 20 starts like this:

20 42 51 65 72 92 114 123 137 144 164 186 195 ...

V  I  N  G  T  V   I   N   G   T   V   I ...

And in the end we have:

...22914648 22914668 22914690 22914699 22914713 22914720 <- hit!

...   T        V        I        N        G

TRENTE (30) is ok, and TRENTE-QUATRE (34) too...

So 5, 20, 30, 34, ... start a sequence of French Self-Belgian Word Numbers. How would look the English Self-Belgian Word Numbers sequence?

(Jens Voss found the first integers under 100: 24, 28, 43, 45, 54, 70)

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[copyleft: E. Angelini, Brussels, Belgium – june 15th, 2005.]

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