First differences of odd rank

[June 7th, 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:

... 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!