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

Fla*Even  would 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, ....

 

[Éric 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. I’m only just past 324000 terms (so far) when I decided to have a look:

 

FlagpoleVariances2

 

It appears that a correspondence search for the letter ‘L’ 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

 

[I’ve 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.]

...

I’ve taken this file down, because I will replace it with something much better. I’m going to do an actual flagpole down to this region, and beyond. In order to keep file-sizes relatively small, I’m 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. There’s lots of white space (which adds to the file-size) but I’m 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 (I’m currently at 340000) region where we encounter the L-spur (spur = railroad terminology). That’s 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 pole’s "eleven-" and "twelve-" thousands are used up and we approach the 411- and 412-thousand Ls for search finds. In other words, I’m 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. I’m up to 600000.

http://chesswanks.com/num/flagpole

 

The next zero variance [a(n) = n] after the one at 321947 is 411273. I’ve 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 don’t think it will look like the L-spur because it will lack the gradual approach to the plateau.

 

I’m 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. I’m also attaching a blow-up of the L-spur region.

 

 

 

 

                                                      Blow up:

 

__________

 

Many thanks to you, Hans and Claudio!

Best,

É.

(August 14th 2012, Brussels, Belgium)