**Runs**

(which self-describe their sizes)

A
few sequences, based on the same idea, which are not in the OEIS at the present
moment (Augustus 24^{th}, 2010)

The
sequence **P** is made of successive
runs of increasing integers (for instance 1,4,7). A run
is completed when the next integer is too small to fit (example 1,4,7,6 – this
6 is not part of the run; thus the run has only three elements, its size is 3).
Last remark: a run can be of size 1 (it then has only one element; according to
the rule, this single element can not be seen as part of the previous run nor
the next one):

**P** = 2, 3, 1, 4, 7, 6,
5, 8, 9, 11, 10, 12, 13, 14, 15, 16, 18, 17, 19, 20, 21, 22, 24, 23, 25, 26,
27, 29, 28, 30, 31, 32, 33, 34, 35, 37, 36, 38, 39, 40, 41, 42, 43, 44, 46, 45,
47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 56, ...

This sequence
shows two nice things:

- first, all natural numbers are present, and there are no
duplicates. P is a permutation of N.

- second, the successive sizes of the runs re-design P itself!

The second point
is clearly visible below.

First we
copy/paste **P**:

**P** = 2, 3, 1, 4, 7, 6, 5, 8, 9, 11, 10, 12, 13, 14, 15, 16, 18, 17, 19, 20, 21,
22, 24, 23, 25, 26, 27, 29, 28, 30, 31, 32, 33, 34, 35, 37, 36, 38, 39, 40, 41,
42, 43, 44, 46, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 56, ...

Then we
systematically insert a star *
between two integers, when the second integer is smaller than the previous one:

**P** = 2, 3,*1, 4, 7,*6,*5, 8, 9, 11,*10, 12, 13, 14, 15, 16,
18,*17, 19, 20, 21,
22, 24,*23, 25, 26,
27, 29,*28, 30, 31,
32, 33, 34, 35, 37,*36,
38, 39, 40, 41, 42, 43, 44, 46,*45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57,*56, ...

We now count the
quantity of integers which are placed between two stars; we get (in red):

**P** = __2, 3__,*__1, 4, 7__,*__6__,*__5, 8, 9, 11__,*__10, 12, 13, 14, 15,
16, 18__,*__17,
19, 20, 21, 22, 24__,*__23,
25, 26, 27, 29__,* ...

2 3 1 4 7 6 5 ...

We see that the
succession of the red figures re-design **P**
itself: 2, 3, 1, 4, 7, 6, 5, ...

**P** is easy to self-construct, as one will quickly
understand, looking at the successive chunks of **P**.

----------

The sequence **Q** deals with the same idea -- but we
handle here *decreasing* runs of
integers:

**Q** = 2, 1, 3, 6, 5,
4, 12, 11, 10, 9, 8, 7, 17, 16, 15, 14, 13, 21, 20, 19, 18, 33, 32, 31, 30, 29,
28, 27, 26, 25, 24, 23, 22, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 54, 53,
52, 51, 50, 49, 48, 47, 46, 45, ...

Again, by
construction, all natural numbers are present with no duplicates.

To see if **Q** correctly self-describes its own
runs, we apply the *star* technique (we
insert a star when we notice that the second integer is *larger* than the previous one):

**Q** = 2, 1,*3,*6, 5, 4,*12, 11, 10, 9, 8, 7,*17, 16, 15, 14, 13,*21, 20, 19, 18,*33, 32, 31, 30, 29, 28,
27, 26, 25, 24, 23, 22,*44,
43, 42, 41, 40, 39, 38, 37, 36, 35, 34,*54, 53, 52, 51, 50, 49, 48, 47, 46, 45,*63, ...

We measure the
size of the runs:

**Q** = __2, 1__,*__3__,*__6, 5, 4__,*__12, 11, 10, 9, 8, 7__,*__17, 16, 15, 14, 13__,*__21, 20, 19, 18__,*__33, 32, 31, 30, 29,
28, 27, 26__, ...

2 1 3
6 5 4 12 ...

Indeed, we get **Q** again (in red)...

----------

To produce more
sequences of this type, we only need our imagination! We must find a suitable
characteristic to define a *run*,
and... let it go!

Some ideas to
define a run:

- a *p-run* is a
sub-sequence (made of consecutive terms) which contains only *one* prime;

- a *c-run* is a
sub-sequence (made of consecutive terms) which contains only *one* non-prime;

- a *t-run* is a
sub-sequence (made of consecutive terms) which contains only *one* triangular number;

- an *o-run* is a
sub-sequence (made of consecutive terms) which contains only *one* odd integer;

- an *e-run* is a
sub-sequence (made of consecutive terms) which contains only *one* even integer;

...

Let’s
try to use the first definition (the *p-run*)
in order to build a sequence of the same type as above (that is: a permutation
of N and a self-description);

Can
we start with “1”? No, this would mean that the first run, which contains a
single prime, is of size 1 – but this 1 is *not*
a prime... We then try to start with 2 – and the rest comes smoothly!

**R** = 2, 1, 3, 5, 4, 6, 7,
8, 9, 10, 12, 11, 14, 15, 16, 13, 18, 20, 21, 22, 24, 17, 25, 26, 27, 28, 30,
32, 19, 33, 34, 35, 36, 38, 39, 40, 23, 42, 44, 45, 46, 48, 49, 50, 51, 29, 54,
55, 56, 57, 58, 60, 62, 63, ...

This
is (again, by construction) a permutation of N. To check if **R** does self-describe its *p-runs* sizes, we will replace the *star* *technique*
with a simpler one: the *carriage-return*
technique!

First
we copy/paste **R**:

**R** = 2, 1, 3, 5, 4, 6,
7, 8, 9, 10, 12, 11, 14, 15, 16, 13, 18, 20, 21, 22, 24, 17, 25, 26, 27, 28,
30, 32, 19, 33, 34, 35, 36, 38, 39, 40, 23, 42, 44, 45, 46, 48, 49, 50, 51, 29,
54, 55, 56, 57, 58, 60, 62, 63, ...

We
then push “carriage return” immediately BEFORE every prime of **R**, starting from the left; we get:

**R** =

2,
1,

3,

5,
4, 6,

7,
8, 9, 10, 12,

11,
14, 15, 16,

13,
18, 20, 21, 22, 24,

17,
25, 26, 27, 28, 30, 32,

19,
33, 34, 35, 36, 38, 39, 40,

23,
42, 44, 45, 46, 48, 49, 50, 51,

29,
52, 54, 55, 56, 57, 58, 60, 62, 63, ...

(We see that the first column is the succession of the primes. And we
understand that the length of each line is dictated by the reading, left to
right and top to bottom, of the elements of the array).

We
now measure the quantity of integers in every line (in red):

**R** =

2 2, 1,

1 3,

3 5, 4, 6,

5 7, 8, 9, 10, 12,

4 11, 14, 15, 16,

6 13, 18, 20, 21, 22, 24,

7 17, 25, 26, 27, 28, 30, 32,

8 19, 33, 34, 35, 36, 38, 39, 40,

9 23, 42, 44, 45, 46, 48, 49, 50,
51,

10 29, 52 54, 55, 56, 57, 58, 60, 62, 63, ...

Yes,
the red figures re-design **R**.

----------

Here
is **S**, where the runs contain only
one odd number:

**S** = 1, 3, 2, 4, 5, 6, 7, 8, 10, 12, 9, 14, 16, 18, 20,
11, 22, 24, 26, 28, 30, 13, 32, 34, 36, 38, 40, 42, 15, 44, 46, 48, 50, 52, 54,
56, 17, 58, 60, 62, 64, 66, 68, 70, 72, 74, 19, 76, 78, ...

*Carriage-return*
check:

**S** =

1,

3, 2, 4,

5, 6,

7, 8, 10, 12,

9, 14, 16, 18, 20,

11, 22, 24, 26,
28, 30,

13, 32, 34, 36,
38, 40, 42,

15, 44, 46, 48,
50, 52, 54, 56,

17, 58, 60, 62,
64, 66, 68, 70, 72, 74, ,

19, 76, 78, ...

Measure
of the *o-runs* (in red):

**S** =

1 1,

3 3, 2, 4,

2 5, 6,

4 7, 8, 10, 12,

5 9, 14, 16, 18, 20,

6 11, 22, 24, 26, 28, 30,

7 13, 32, 34, 36, 38, 40, 42,

8 15, 44, 46, 48, 50, 52, 54, 56,

10 17, 58, 60, 62, 64, 66, 68, 70, 72, 74,

12 19, 76, 78, ...

----------

Here
is **T**, where the *e-runs* must contain only one *even*
number (a quick check is done by underlining the runs):

**T** = __2, 1__, __4__, __6,
3, 5, 7__, __8, 9, 11, 13, 15, 17__, __10, 19, 21__, __12, 23, 25, 27, 29,__ __14, 31, 33, 35, 37, 39, 41__, ...

2 1 4
6 3 5 7

Hope
this technique is now clear – and will be used by others!

__________

Latest
news (September 8^{th}, 2010), by **Neil
Sloane**:

>
Eric, I'm adding
these 5 seqs to the OEIS - see A171083-A171087

Thank
you, Neil!

Best,

É.