**First differences of odd
rank**

[June
7^{th}, 2010]

Hello
SeqFans, [http://list.seqfan.eu]

I’ve
had the idea of a self-describing seq:

«The
first differences of odd rank are the sequence S itself»

If S must be monotonically
increasing we get:

S = 1 2 3 5 6 9 10 15 16 22 23 32 33 43 44
59 60 76 77 99 100
123 ...

1st
diff: 1 1 2 1 3 1
5 1 6
1 9 1
10 1 15
1 16 1
22 1 23

odd diff: 1 . 2 . 3 . 5 . 6 .
9 . 10 .
15 . 16 .
22 . 23 .

... as one can see, odd rank diff are S itself.

Now
the difficult part:

After
seing S, one could wonder if, dropping the increasing constraint, it
would be possible for T
to be a permutation of the Naturals (1,2,3,4,5,6,7,8,..., n)

I
think it is possible, yes -- but T is a nightmare to construct:

T = 1 2 3 5 4 7 6 11 8 12 9 16 13 19 10 21 14 22 15 27 17
26 18 34 ...

1st diff: 1 1 2 1 3 1 5 3 4 3
7 3
6 9 11
7 8 7
12 10 9
8 16

odd diff: 1 . 2 . 3 . 5 . 4 . 7 .
6 . 11 .
8 . 12
. 9 .
16

... as one can see again, odd rank diff are T itself -- and no term is
a copy of a previous one (first missing terms are 20,23,24,25,28,...)

But
how was T constructed?
(I hope I made no mistake)

As
always, I’ve wanted the new (free) term of T to be *the
smallest term not yet present in T*; but one has to be careful, see:

T = 1 2 3 5 4 7 6 11 8 12 9 16

1st
diff: 1 1 2 1 3 1 5 3 4
3 7

odd diff: 1 . 2 . 3 . 5
. 4 .
7

... now we see that the *smallest
term not yet present in T*
could be 10:

T = 1 2 3 5 4 7 6 11 8 12 9 16 10

1st diff: 1 1 2 1 3 1 5 3 4
3 7 6

odd diff: 1 . 2 . 3 . 5 . 4 . 7 .

But
10 leads to an impossibility because of the "odd diff" line which
imposes "6" as the next "first diff":

T = 1 2 3 5 4 7 6 11 8 12 9 16 10

1st
diff: 1 1 2 1 3 1 5 3 4
3 7 6 6

odd diff: 1 . 2 . 3 . 5
. 4 .
7 . 6

... and we have no available integer to expand T: 10+6=16 -- and 16 is already
taken

or 10-6= 4 -- and 4
is already taken...

So
we must discard 10 and try the next *smallest
term not yet present in T*
which is "13":

T = 1 2 3 5 4 7 6 11 8 12 9 16 13 19 a
b
c d e
f g h i j ...

1st
diff: 1 1 2 1 3 1 5 3 4
3 7 3 6
? 11 ?
8 ? 12
? 9 ?
16

odd diff: 1 . 2 . 3 . 5
. 4 .
7 . 6 .
11 . 8 .
12 . 9 .
16

If
this is of interest, could someone have a try on S and T?

Best,

É.

[I’ve
forgotten to mention this link:

http://www.research.att.com/~njas/sequences/A033485

... where both the odd ranks of the first diff and the even
ranks are the seq itself.]

_______________

**Douglas McNeil** was
the first to answer:

> «The first
differences of odd rank are the sequence S itself»

> [...]

>
S = 1 2 3 5 6 9 10
15 16 22 23 32 33 43 44 59 60 76 77
99 100 123 ...

I agree with your
terms:

sage: S

[1, 2, 3, 5, 6, 9,
10, 15, 16, 22, 23, 32, 33, 43, 44, 59, 60, 76, 77, 99, 100, 123, 124, 156,
157, 190, 191, 234, 235, 279, 280, 339, 340, 400, 401, 477, 478, 555, 556, 655,
656, 756, 757, 880, 881, 1005, 1006, 1162, 1163, 1320, 1321, 1511, 1512, 1703,
1704, 1938, 1939, 2174, 2175, 2454, 2455, 2735, 2736, 3075, 3076, 3416, 3417,
3817, 3818, 4219, 4220, 4697, 4698, 5176, 5177, 5732, 5733, 6289, 6290, 6945,
6946, 7602, 7603]

> After seing S
one could wonder if, dropping the increasing constraint, it would be possible
for T to be a permutation of the Naturals (1,2,3,4,5,6,7,8,..., n)

>

> I think it is
possible, yes -- but T
is a nightmare to construct:

>

>
T = 1 2 3 5 4 7 6
11 8 12 9 16 13 19 10 21 14 22 15 27
17 26 18 34 ...

I
agree with T too--
although I can never tell with these sequences whether my construction matches
yours:

sage: T

[1,
2, 3, 5, 4, 7, 6, 11, 8, 12, 9, 16, 13, 19, 10, 21, 14, 22, 15, 27, 17, 26, 18,
34, 20, 33, 23, 42, 25, 35, 24, 45, 29, 43, 28, 50, 31, 46, 30, 57, 32, 49, 36,
62, 37, 55, 38, 72, 39, 59, 40, 73, 41, 64, 44, 86, 51, 76, 47, 82, 53, 77, 48,
93, 52, 81, 54, 97, 56, 84, 58, 108, 60, 91, 61, 107, 65, 95, 63, 120, 66, 98,
67, 116, 68, 104, 69, 131, 74, 111, 70, 125, 71, 109, 75, 147, 78, 117, 79,
138, 83, 123, 80, 153, 85, 126, 87, 151, 88, 132, 89, 175, 90, 141, 92, 168, 96,
143, 94, 176, 99, 152, 100, 177, 101, 149, 102, 195, 103, 155, 105, 186, 106, 160,
110, 207, 113, 169, 112, 196, 114, 172, 115, 223, 118, 178, 119, 210, 121, 182,
122, 229, 124, 189, 127, 222, 128, 191, 129, 249, 133, 199, 130, 228, 134, 201,
135, 251, 136, 204, 137, 241, 139, 208, 140, 271, 142, 216, 144, 255, 145, 215,
148, 273, 146, 217, 150, 259, 156, 231, 154, 301, 157, 235, 158, 275, 159, 238,
161, 299, 162, 245, 163, 286, 164, 244, 165, 318, 167, 252, 166, 292, 170, 257,
171, 322, 173, 261, 174, 306, 179, 268, 180, 355, 184, 274, 183, 324, 185, 277,
181, 349, 187, 283, 188, 331, 190, 284, 192, 368, 194, 293, 193, 345, 197, 297,
198, 375, 202, 303, 203, 352, 200, 302, 205, 400, 206, 309, 209, 364, 211, 316,
212, 398, 213, 319, 214, 374, 218, 328, 219, 426, 220, 333, 221, 390, 224, 336,
225, 421, 226, 340, 227, 399, 232, 347, 230, 453, 233, 351, 234, 412, 237, 356,
236, 446, 239, 360, 240, 422, 243, 365, 242, 471, 246, 370, 247, 436, 250, 377,
248, 470, 253, 381, 254, 445, 256, 385, 258, 507, 260, 393, 262, 461, 264, 394,
263, 491, 267]

Doug

--

Department of
Earth Sciences

University of Hong
Kong

_______________

Many
thanks again, **Doug**, this is perfect!