The same sequence, but differently

(Dall-e creation)

We will look today at sequences whose terms can be modified without this modification affecting the sequence itself.
It seems paradoxical, but look instead:

a) Add 9 to each digit of A, then follow the result with a comma; A does not change:

A = 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, 10, 15, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 14, 10, 9, 18, 10, 17,...

Check:
– the 1st digit of A is 1
– we add 9 to 1 and get 10 [which is indeed a(1)]
– the 2nd digit of A is 0
– we add 9 to 0 and get 9 [which is indeed a(2)]
– the 3rd digit of A is 9
– we add 9 to 9 and get 18 [which is indeed a(3)], etc.

b) Add 18 to each digit of B, then follow the result with a comma; B stays the same:
B = 20, 18, 19, 26, 19, 27, 20, 24, 19, 27, 20, 25, 20, 18, 20, 22, 19, 27, 20,...

c) Add 27 to each digit of C, then follow the result with a comma; C does not change:
C = 30, 27, 29, 34, 29, 36, 30, 31, 29, 36,...

d) Add 36 to each digit of D, then follow the result with a comma; D stays the same:
D = 40, 36, 39, 42, ...
Etc.

What about multiplying instead od adding? Multiplying each digit by 10 or 11 is a bit dull – using 12 is more interesting:

M = 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 108, 72, 24, 48, 48, 96, 48, 96, 108, 72, 48, 96, 108, 72, 12, 0, 96, 84, 24, 24, 48, ...
Check:
– the 1st digit of M is 1
– we multiply 1 by 12 and get 12 [which is indeed m(1)]
– the 2nd digit of M is 2
– we multiply 2 by 12 and get 24 [which is indeed m(2)]
– the 3rd digit of M is 2
– we multiply 2 by 12 and get 24 [which is indeed m(3)], etc.
___________________
Next day update (Jan 25th 2024)
Giorgos Kalogeropoulos almost immediately corrected and extended the sequence A, with a commentary:
> Here are the first 100 terms

A = 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, 10, 15, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 14, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 13, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, 10, 15, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 17, 10, 9, 18, 10, 9, 10, 12, 10, 9, 18, 10, 17, 10, 9, 10, 16, 10, 9, 18, 10, 9, ...

> The only integers that appear in the sequence are [9, 10, 11, 12, 13, 14, 15, 16, 17, 18].
Here are the number of times each of those numbers appears in the first 100000 terms:
{10, 43146},
{9, 24597},
{18, 14022},
{17, 7996},
{16, 4559},
{15, 2598}, 
{14, 1481},
{13, 844},
{12, 482},
{11, 275}

> The index (positions) of these numbers in the seq are:
9 ~~> 2, 7, 11, 14, 18, 23, 26, 30, 35, 39, 44, 47, 51, 56, 60, 63, 67, 72, 76, 81 ... OEIS A287107
10 ~~> 1, 4, 6, 8, 10, 13, 15, 17, 20, 22, 25, 27, 29, 32, 34, 36, 38,  41, 43, 46 ... OEIS  A287106
11 ~~> 151, 616, 966, 1231, 1580, 2045, 2310, 2658, 3123, 3473, 3938, 4203, 4549, 5014, 5364, 5629, 5978, 6443, 6793, 7258 ... not in OEIS
12 ~~> 86, 351, 551, 702, 901, 1166, 1317, 1515, 1780, 1980, 2245, 2396, 2593, 2858, 3058, 3209, 3408, 3673, 3873, 4138 ... not in OEIS
13 ~~> 49, 200, 314, 400, 514, 665, 751, 864, 1015, 1129, 1280, 1366, 1478, 1629, 1743, 1829, 1943, 2094, 2208, 2359 ... not in OEIS
14 ~~> 28, 114, 179, 228, 293, 379, 428, 493, 579, 644, 730, 779, 843, 929, 994, 1043, 1108, 1194, 1259, 1345 ... not in OEIS
15 ~~> 16, 65, 102, 130, 167, 216, 244, 281, 330, 367, 416, 444, 481, 530, 567, 595, 632, 681, 718, 767 ... not in OEIS
16 ~~> 9, 37, 58, 74, 95, 123, 139, 160, 188, 209, 237, 253, 274, 302, 323, 339, 360, 388, 409, 437 ... not in OEIS
17 ~~> 5, 21, 33, 42, 54, 70, 79, 91, 107, 119, 135, 144, 156, 172, 184, 193, 205, 221, 233, 249 ... not in OEIS
18 ~~> 3, 12, 19, 24, 31, 40, 45, 52, 61, 68, 77, 82, 89, 98, 105, 110, 117, 126, 133, 142 ... not in OEIS

> As it is difficult to search for patterns in this seq, here is a visualization of the first 10000 terms cut in 100 chunks of 100 terms:

> As we can see, although we can find repeating subsequences of any length, the whole pattern is not repeating.

> When we  "add x to every digit" we can have at most 10 results and no other, namely x+0, x+1,... x+9
> So, in order to get more results we can use a new rule like:
Instead of having a fixed x, for every term a(n) of the seq we can add a(n) to every digit of a(n). 
Starting with a(1)=1 the sequence goes like: 
{1}, 
{2},
{4},  
{8}, next step 8+8=16
{16}, next step 16+1=17 and 16+6=22
{17, 22}, next step 17+1=18, 17+7=24, 22+2=24, 22+2=24   
{18, 24, 24, 24}, 
{19, 26, 26, 28, 26, 28, 26, 28},
{20, 28, 28, 32, 28, 32, 30, 36, 28, 32, 30, 36, 28, 32, 30, 36}, 
{22, 20, 30, 36, 30, 36, 35, 34, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42}, 
{24, 24, 22, 20, 33, 30, 39, 42, 33, 30, 39, 42, 38, 40, 37, 38, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44}

> The final sequence is

F = 1, 2, 4, 8, 16, 17, 22, 18, 24, 24, 24, 19, 26, 26, 28, 26, 28, 26, 28, 20, 28, 28, 32, 28, 32, 30, 36, 28, 32, 30, 36, 28, 32, 30, 36, 22, 20, 30, 36, 30, 36, 35, 34, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42, 30, 36, 35, 34, 33, 30, 39, 42, 24, 24, 22, 20, 33, 30, 39, 42, 33, 30, 39, 42, 38, 40, 37, 38, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, 33, 30, 39, 42, 38, 40, 37, 38, 36, 36, 33, 30, 42, 48, 46, 44, ...
... which is not in OEIS

> F has also a fractal plot, first 4000 terms hereunder:
> First 133000 terms:
ÉA
This is just brilliant, Giorgos – many thanks!
____________________
Jan. 27th update

ÉA to GK
> Hi Giorgos (...) you explained to me that the only integers that appear in the seq A opening this page are [9, 10, 11, 12, 13, 14, 15, 16, 17 and 18]. This made me think: what about the seq T and (another/different) new rule: 
« Add 9 to the 1st digit of T, 10 to the 2nd digit of T, 11 to the 3rd digit, 12 to the 4th, 13 to the 5th, etc. so that the seq T will be unchanged ».

T = 10, 10, 12, 12, 14, 16, 16, 18, 18, 22, 20, 26, 22, 28, 24, 32, 26, 34, 29, ... »

> I guess that the number of distinct integers is not limited anymore to 10 here — what do you think? does the above T work?

GK
> Yes, it works; the first 100 terms are:

 T = 10, 10, 12, 12, 14, 16, 16, 18, 18, 22, 20, 26, 22, 28, 24, 32, 26, 34, 29, 30, 31, 30, 33, 38, 35, 36, 37, 44, 39, 42, 42, 42, 43, 48, 46, 48, 47, 55, 50, 48, 52, 51, 54, 52, 56, 57, 58, 64, 60, 63, 62, 66, 64, 69, 67, 68, 68, 75, 71, 70, 73, 72, 75, 74, 77, 77, 79, 84, 81, 84, 83, 88, 85, 89, 88, 89, 90, 86, 91, 96, 94, 92, 96, 93, 98, 98, 100, 98, 102, 104, 104, 107, 106, 110, 109, 108, 111, 106, 113, 111 > Here is a 100 terms plot:
> The 1000 terms plot is:
ÉA
Wonderful Giorgos – I will submit both seq A and seq T to the OEIS in a couple of hours – many thanks!

(Dall-e creation)
(Boomerangs updates so far)








Commentaires

Enregistrer un commentaire

Posts les plus consultés de ce blog

Beautés ?

Image
Beau comme un clavier ? Un deuxième ? (celui de Modotti)   Beau comme le dos d ’ un timbre proposé par Delcampe ? Beau comme une série ukrainienne où les avions râpent le Kremlin ? Belle comme Chargois ? ... belle comme Germaine Richier ? Bel atelier ? Beau comme une pièce de 1 santim ? ... ou comme une pièce de 50  santeem ? ... Byrrh , beau comme un anus ? Beau comme la collection de Marin Karmitz à Beaubourg ? Belles Ménines ? Beau comme Marcel Duchamp ? Beau comme Jean-Luc Godard ? Beau comme la réouverture de l ’ Einsteinturm à Berlin ? Beau comme un anneau dudit ? Beau comme Eisenstein ? Beau comme les fleurs de Pascale hier matin ? Beau comme on dirait la planète Mars ? Beau comme des recherches personnelles ? Beau comme une marelle (vers le ciel) ? Beau comme un triple selfie (au Man Ray ) ? Beau comme le ciel de Sacha , mardi, en face du restaurant La Montagne ? (qui a rouvert) ... beau comme ce bleu-là de Lo ? Beau comme le froid ? Beau comme une Vespa ? Beau comme Cloon
Lire la suite

Underline, reproduce

Image
Dall.e creation If we underline the... « first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, tenth, eleventh,… » letter of this sentence S 1 , we see that the successive underlined letters simply reproduce S 1 . No big deal. If we decide not to start S 1   with "first" but with "sixth", we can reproduce the new S 6  quite easily: S 6 = sixth , second, third, fourth, fifth, first , seventh, eighth, ninth, tenth, eleventh,… S 6  was reproduced in a slightly less mechanical manner than  S 1 :  just two words of  S 1   swapped here ("first" and "sixth", in red, above). If we start S 8  with "eighth" we just swap again a pair of words of S 1  ("first" and "eighth", in red, hereunder): S 8  = eighth , second, third, fourth, fifth, sixth, seventh, first , ninth, tenth, eleventh,… What about starting S 9  with "ninth"? How many swaps do we need to reproduce S 9 ?  This is more tricky! (Indeed, we
Lire la suite

Le tripalin se présente

Image
Un tripalin est constitué de trois images. Chaque image illustre un substantif. Accolés, ces trois substantifs forment une chaîne palindromique. Laquelle nous vous invitons à trouver.  Exemple ci-dessous. Le premier mot est COL-BLEU Le second DUEL Le troisième BLOC . Ces mots forment la chaîne COLBLEUDUELBLOC qui est bien palindromique. Je posterai chaque jeudi une nouvelle salve de tripalins . Rien que des noms illustrés – pas d’adjectifs, pas de verbes conjugués, pas d’adverbes (parfois un prénom ou un nom propre – mais seulement quand il faut). À la demande générale (de mon neveu), chaque tripalin sera surmonté de son numéro d ’ ordre ( octothorpe noir) et de trois nombres bleus. Ces nombres reprennent les trois longueurs (en lettres) des trois mots présents dans le tripalin. On a par exemple ci-dessous [ #0 ↓ 7+4+4 ] car COL-BLEU comporte  7  lettres, DUEL   4  et BLOC   4 . Le numéro d ’ ordre est zéro. #0 ↓ 7+4+4 Voilà – tous retours critiques, construits ou déconstruits, bien
Lire la suite