Flagpole

of themselves (sequences)

 

Hello SeqFans,

a little word sequence not (yet?) in the OEIS:

 

1††††††††††††††††††††††††† O N E

7††††††††††††††††† S E V E N

3††††††††††††††††††† T H R E E

6††††††††††††† ††††††††††††S I X

5††††††††††††††††††† F I V E

11††††††††††††††††††† E L E V E N

8††††††††††††††††††††††††† E I G H T

9††††††††††††††††††††††††† N I N E

2††††††††††††††††††††††††† T W O

13††††††††††††††††††††††† T H I R T E E N

4†††††††††††††† †††††F O U R

10††††††††††††††††††††††† T E N

12††††††††††††††††††††† T W E L V E

16††††††††††††††††††††††††† S I X T E E N

15††††††††††††††††††††††† F I F T E E N

26††††††††† T W E N T Y S I X

14††††††††††††††††††††††††† F O U R T E E N

18††††††††† ††††††††††††††E I G H T E E N

17††††††††††††††††††††† S E V E N T E E N

19††††††††††††††††††† N I N E T E E N

20††††††††††††††††††††† T W E N T Y

111††† O N E H U N D R E D E L E V E N

21††††††††††††††††††††† T W E N T Y O N E

25††††††††† T W E N T Y F I V E

22††††††††††††††††††††† T W E N T Y T W O

23††††††††††††††††††† T W E N T Y T H R E E

24††††††††††††††††††††† T W E N T Y F O U R

28††††††††††† T W E N T Y E I G H T

38††††††††† T H I R T Y E I G H T

30††††††††††††††††††††††† T H I R T Y

27††††††††††††††††††††††††† T W E N T Y S E V E N

29††††††††††††††††††† T W E N T Y N I N E

...†††††††††††††††††††††††† I

†††††††††††††††††††††††††††† N

†††††††††††††††††††††††††††† E

†††††††††††††††††††††††††††† T

†††††††††††††††††††††††††††† W

†††††††††††††††††††††††††††† O

††††††††††††††††††††††††††††

 

Explanation: the (yellow) "single letter column" (the "flagpole"), spells vertically O N E S E V E N ... which is the sequence itself. In extending the sequence, always try to use the smallest Natural not yet present in the sequence.

 

Is the sequence a permutation of the Naturals?

 

 

[Hans Havermann]:

 

The terms of this sequence are pretty close to a straight line. In fact, of the first 10000 terms, I count 5832 where a(n) = n, beginning with n = 1, 3, 5, 15, 18, 28, 30, 45, 65, 113, ..., and the variances from this line for the rest is (in this range) relatively small:

 

 

FlagpoleVariances

 

[Eric Angelini]:

 

Below is the same idea as above -- but with integers instead (the yellow "digit column b" spells the digits of the sequence):

 

Digit column:a b c

 

††††††††††††††††† 1

††††††††††††††††† 2

††††††††††††††††† 3

††††††††††††††††† 4

††††††††††††††††† 5

†††††††††††††† †††6

††††††††††††††††† 7

††††††††††††††††† 8

††††††††††††††††† 9

††††††††††††††††† 1 0

††††††††††††††† 2 0

††††††††††††††† 1 2

††††††††††††††† 3 0

††††††††††††††††† 1 1

††††††††††††††††† 2 1

††††††††††††††† 1 3

††††††††††††††† 4 0

††††††††††††††††† 1 4

†† †††††††††††††††1 5

††††††††††††††††† 2 2

††††††††††††††††† 1 6

††††††††††††††††† 1 7

††††††††††††††† 2 3

††††††††††††††† 2 4

††††††††††††††† 5 0

††††††††††††††††† 1 8

††††††††††††††† 3 4

††††††††††††††††† 1 9

††††††††††††††† 2 5

††††††††††††††††† 2 6

†††† †††††††††††††2 7

††††††††††††††††† .

††††††††††††††††† .

††††††††††††††††† .

 

The above sequence 1,2,3,4,5,6,7,8,9,10,20,12,30,11,21,13,40,14,15, 22,16,17,23,24,50,18,34,19,25,26,27,... is for sure a permutation of the Naturals.

 

And if this "flagpole" idea is worth entering the OEIS, then one could add a lot of such sequences:

Fla*Prime would start: 2,3,5,7,11,13,17,23,19,37,29,31,41,59,37,47,...

Fla*Fibowould start: 0,1,1,2,3,5,8,13,34,233,144,21,46368,121393,...

Fla*Evenwould start: 2,4,6,8,10,20,12,30,14,22,32,40,16,24,26,28,...

 

 

[Claudio Meller]:

 

Fla in Spanish: 1, 5, 2, 4, 6, 9, 8, 11, 10, 12, 3, 13, 15, 14, 7, 23, 18, 16, 17, 19, 22, 20, 24, 25, 26, ....

 

[…ric Angelini]:

 

Fla in French: 1, 5, 25, 3, 9, 4, 20, 6, 11, 21, 7, 35, 8, 15, 14, 13, 23, 12, 10, 16, 19, 2, 18, 29, 24, 22, 34, 17, 30, 27, 26, 28, 31, 80, 32, 33, 36, 42, 43, 37, 71, 38, ...

 

 

††††††††††††††††††††††† U N

††††††††††††††††††† C I N Q

††††††††††††† V I N G T-C I N Q

†††††††††††† †††††T R O I S

††††††††††††††††††††††† N E U F

††††††††††††††††††††††† Q U A T R E

††††††††††††††††††††††† V I N G T

††††††††††††††††††††† S I X

††††††††††††††††††††† O N Z E

††††††††††††††††† V I N G T E T U N

††††††††††††††††† S E P T

††††††††††† T R E N T E-C I N Q

††††††††††††††††††† H U I T

††††††††††††††††† Q U I N Z E

††††††††††††††††††††††† Q U A T O R Z E

††††††††††††††††††††††† T R E I Z E

††††††††††† V I N G T-T R O I S

††††††††††††††††††††† D O U Z E

††††††††††††††††††††† D I X

†††††††††††††††† †††††††S E I Z E

††††††††††††††††† D I X-N E U F

††††††††††††††††††††† D E U X

††††††††††††††† D I X-H U I T

††††††† V I N G T-N E U F

††††††††††††† V I N G T-Q U A T R E

††††††††† V I N G T-D E U X

††††††† T R E N T E-Q U A T R E

††††††††††† D I X-S E P T

††††††††††††††††††††† T R E N T E

††††††††††† V I N G T-S E P T

††††††††††††††††††††††† V I N G T-S I X

††††††††††††††††††††† V I N G T-H U I T

††††††††††††††††† T R E N T E E T U N

††††† Q U A T R E-V I N G T S

††††††††††††††††††††††† T R E N T E-D E U X

††† T R E N T E-T R O I S

††††††††† T R E N T E-S I X

Q U A R A N T E-D E U X

††† Q U A R A N T E-T R O I S

††††††††††††††††† T R E N T E-S E P T

S O I X A N T E E T O N Z E

††††††††††††††††††† T R E N T E-H U I T

†††††††††††††††††††††† ...

 

[Hans Havermann]:

 

I had a hunch that the linear behaviour that I charted might have been a small-numbers-only phenomenon so I decided to calculate one million terms to see if the small variances continued. Iím only just past 324000 terms (so far) when I decided to have a look:

 

FlagpoleVariances2

 

It appears that a correspondence search for the letter ĎLí is responsible for the increasing positive variances starting with term 321680:

 

321680 +†† 31†† three hundred twenty-one thousand seven hundred eleven

321859 +†† 52†† three hundred twenty-one thousand nine hundred eleven

321873 +†† 39†† three hundred twenty-one thousand nine hundred twelve

321890 +121†† three hundred twenty-two thousand eleven

321909 +103†† three hundred twenty-two thousand twelve

321927 +184†† three hundred twenty-two thousand one hundred eleven

321945 +167†† three hundred twenty-two thousand one hundred twelve

321962 +249†† three hundred twenty-two thousand two hundred eleven

321983 +229†† three hundred twenty-two thousand two hundred twelve

322002 +309†† three hundred twenty-two thousand three hundred eleven

322021 +291†† three hundred twenty-two thousand three hundred twelve

322039 +372†† three hundred twenty-two thousand four hundred eleven

322056 +356†† three hundred twenty-two thousand four hundred twelve

322078 +433†† three hundred twenty-two thousand five hundred eleven

322092 +420†† three hundred twenty-two thousand five hundred twelve

322098 +513†† three hundred twenty-two thousand six hundred eleven

322114 +498†† three hundred twenty-two thousand six hundred twelve

322118 +593†† three hundred twenty-two thousand seven hundred eleven

322140 +572†† three hundred twenty-two thousand seven hundred twelve

322161 +650†† three hundred twenty-two thousand eight hundred eleven

322183 +629†† three hundred twenty-two thousand eight hundred twelve

322208 +703†† three hundred twenty-two thousand nine hundred eleven

322231 +681†† three hundred twenty-two thousand nine hundred twelve

322251 +760†† three hundred twenty-three thousand eleven

322275 +737†† three hundred twenty-three thousand twelve

322297 +814†† three hundred twenty-three thousand one hundred eleven

322320 +792†† three hundred twenty-three thousand one hundred twelve

322343 +868†† three hundred twenty-three thousand two hundred eleven

322367 +845†† three hundred twenty-three thousand two hundred twelve

322390 +921†† three hundred twenty-three thousand three hundred eleven

322413 +899†† three hundred twenty-three thousand three hundred twelve

322438 +973†† three hundred twenty-three thousand four hundred eleven

322463 +949†† three hundred twenty-three thousand four hundred twelve

322487 + 1024†† three hundred twenty-three thousand five hundred eleven

322507 + 1005†† three hundred twenty-three thousand five hundred twelve

322530 + 1081†† three hundred twenty-three thousand six hundred eleven

322553 + 1059†† three hundred twenty-three thousand six hundred twelve

322578 + 1133†† three hundred twenty-three thousand seven hundred eleven

322602 + 1110†† three hundred twenty-three thousand seven hundred twelve

322624 + 1187†† three hundred twenty-three thousand eight hundred eleven

322648 + 1164†† three hundred twenty-three thousand eight hundred twelve

322673 + 1238†† three hundred twenty-three thousand nine hundred eleven

322698 + 1214†† three hundred twenty-three thousand nine hundred twelve

322722 + 1289†† three hundred twenty-four thousand eleven

322741 + 1271†† three hundred twenty-four thousand twelve

322764 + 1347†† three hundred twenty-four thousand one hundred eleven

322788 + 1324†† three hundred twenty-four thousand one hundred twelve

322810 + 1401†† three hundred twenty-four thousand two hundred eleven

322834 + 1378†† three hundred twenty-four thousand two hundred twelve

322857 + 1454†† three hundred twenty-four thousand three hundred eleven

322879 + 1433†† three hundred twenty-four thousand three hundred twelve

322901 + 1510†† three hundred twenty-four thousand four hundred eleven

322925 + 1487†† three hundred twenty-four thousand four hundred twelve

322948 + 1563†† three hundred twenty-four thousand five hundred eleven

322967 + 1545†† three hundred twenty-four thousand five hundred twelve

322989 + 1622†† three hundred twenty-four thousand six hundred eleven

323013 + 1599†† three hundred twenty-four thousand six hundred twelve

323035 + 1676†† three hundred twenty-four thousand seven hundred eleven

323058 + 1654†† three hundred twenty-four thousand seven hundred twelve

323081 + 1730†† three hundred twenty-four thousand eight hundred eleven

323103 + 1709†† three hundred twenty-four thousand eight hundred twelve

323127 + 1784†† three hundred twenty-four thousand nine hundred eleven

323151 + 1761†† three hundred twenty-four thousand nine hundred twelve

323174 + 1837†† three hundred twenty-five thousand eleven

323193 + 1819†† three hundred twenty-five thousand twelve

323215 + 1896†† three hundred twenty-five thousand one hundred eleven

323239 + 1873†† three hundred twenty-five thousand one hundred twelve

323262 + 1949†† three hundred twenty-five thousand two hundred eleven

323285 + 1927†† three hundred twenty-five thousand two hundred twelve

323307 + 2004†† three hundred twenty-five thousand three hundred eleven

323331 + 1981†† three hundred twenty-five thousand three hundred twelve

323355 + 2056†† three hundred twenty-five thousand four hundred eleven

323378 + 2034†† three hundred twenty-five thousand four hundred twelve

323399 + 2112†† three hundred twenty-five thousand five hundred eleven

323423 + 2089†† three hundred twenty-five thousand five hundred twelve

323447 + 2164†† three hundred twenty-five thousand six hundred eleven

323473 + 2139†† three hundred twenty-five thousand six hundred twelve

323498 + 2213†† three hundred twenty-five thousand seven hundred eleven

323523 + 2189†† three hundred twenty-five thousand seven hundred twelve

323547 + 2264†† three hundred twenty-five thousand eight hundred eleven

323573 + 2239†† three hundred twenty-five thousand eight hundred twelve

323599 + 2312†† three hundred twenty-five thousand nine hundred eleven

323624 + 2288†† three hundred twenty-five thousand nine hundred twelve

323649 + 2362†† three hundred twenty-six thousand eleven

323669 + 2343†† three hundred twenty-six thousand twelve

323692 + 2419†† three hundred twenty-six thousand one hundred eleven

323715 + 2397†† three hundred twenty-six thousand one hundred twelve

323738 + 2473†† three hundred twenty-six thousand two hundred eleven

323762 + 2450†† three hundred twenty-six thousand two hundred twelve

323786 + 2525†† three hundred twenty-six thousand three hundred eleven

323811 + 2501†† three hundred twenty-six thousand three hundred twelve

323836 + 2575†† three hundred twenty-six thousand four hundred eleven

323861 + 2551†† three hundred twenty-six thousand four hundred twelve

323885 + 2626†† three hundred twenty-six thousand five hundred eleven

323905 + 2607†† three hundred twenty-six thousand five hundred twelve

323928 + 2683 ††three hundred twenty-six thousand six hundred eleven

323951 + 2661†† three hundred twenty-six thousand six hundred twelve

323975 + 2736†† three hundred twenty-six thousand seven hundred eleven

323999 + 2713†† three hundred twenty-six thousand seven hundred twelve

 

[Iíve placed a text file of terms 321500 - 324000 (so that you can better appreciate this emergent behaviour) here. It will be interesting to see how this evolves.]

...

Iíve taken this file down, because I will replace it with something much better. Iím going to do an actual flagpole down to this region, and beyond. In order to keep file-sizes relatively small, Iím breaking the pole up into 50000-term pieces, but even with that the first file is 7 MB:

 

http://chesswanks.com/num/flagpole/000050.txt

 

Depending on your browser, you might need to shrink the font-size to keep things from wrapping. Thereís lots of white space (which adds to the file-size) but Iím thinking of extending the flagpole (eventually) into the millions and I expect a lot of that white space to disappear in those larger regions. So why keep the white space in the smaller regions? I want smooth annealing: If you copy/paste (part of) one file into another, I still want everything to line up.

 

You can see which files are available by refreshing:

 

http://chesswanks.com/num/flagpole

 

The program that is actually calculating the terms has really slowed down in this (Iím currently at 340000) region where we encounter the L-spur (spur = railroad terminology). Thatís because my program starts looking at the smallest available number and increments by ones from there. For every L that it now encounters, it has to go many thousands of numbers into the search, which is not very efficient. The good news is that I fully expect things to get back on track (more railroad terminology) once the poleís "eleven-" and "twelve-" thousands are used up and we approach the 411- and 412-thousand Ls for search finds. In other words, Iím guessing that the increasingly large a(n) = n variances generated by the letter L will stabilize at 400000, or so, and disappear soon thereafter.

 

 

[Hans Havermann] (August 24th update):

 

OK. I misjudged how fast the program would run beyond 386700. Iím up to 600000.

http://chesswanks.com/num/flagpole

 

The next zero variance [a(n) = n] after the one at 321947 is 411273. Iíve been thinking about the supply side of specific letters. I had suggested that W might create a spur approaching 700000, because of demand, but I no longer think that. The letters E, F, G, H, I, N, O, R, S, T, U, V, W, and X occur every ten number-words, so there is always plentiful local supply for these, regardless of demand. Y less so, but still plentiful enough. L less than Y but cannot ever be a problem after we reach the -illion numbers. D is easily satisfied once we reach -hundred-, and A, once we reach -thousand- numbers. (Think of how we will supply demand for A after we reach 10^6.) That leaves M after we reach 10^9 (subsequently, B, Q, P, and C, but that is far beyond our scope). Every demand for M after 10^9 will have to travel to 10^9 + 10^6 for its supply, so this should create the next *significant* spur. I donít think it will look like the L-spur because it will lack the gradual approach to the plateau.

 

Iím attaching a graph of the flagpole sequence, to 500000, superimposed on a red a(n) = n line. Most of the red line is not visible, because the flagpole sequence is so close to it. Iím also attaching a blow-up of the L-spur region.

 

 

 

 

††††††††††††††††††††††††††††††††††††††††††††††††††††† Blow up:

 

__________

 

Many thanks to you, Hans and Claudio!

Best,

….

(August 14th 2012, Brussels, Belgium)