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The UUDDUUD sequence, or the 3/7th sequence

Posted 9 years ago

Start with 4. Progressively add or subtract the integers, repeating the sequence + + - - + + -.

FoldList[Plus,4,Flatten[Table[{1,1,-1,-1,1,1,-1}, {20}]] Range[140]]

4, 5, 7, 4, 0, 5, 11, 4, 12, 21, 11, 0, 12, 25, 11, 26, 42, 25, 7, 26, 46, 25, 47, 70, 46, 21, 47, 74, 46, 75, 105, 74, 42, 75, 109, 74, 110, 147, 109, 70, 110, 151, 109, 152, 196, 151, 105, 152, 200, 151, 201, 252, 200, 147, 201, 256, 200, 257, 315, 256, 196, 257, 319, 256, 320, 385, 319, 252, 320, 389, 319, 390, 462, 389, 315, 390, 466, 389, 467, 546, 466, 385, 467, 550, 466, 551, 637, 550, 462, 551, 641, 550, 642, 735, 641, 546, 642, 739, 641, 740, 840, 739, 637, 740, 844, 739, 845, 952, 844, 735, 845, 956, 844, 957, 1071, 956, 840, 957, 1075, 956, 1076, 1197, 1075, 952, 1076, 1201, 1075, 1202, 1330, 1201, 1071, 1202, 1334, 1201, 1335, 1470, 1334, 1197, 1335, 1474, 1334

The differences are 1, 2, -3, -4, 5, 6, -7, .... 8, 9, -10, -11, 12, 13, -14, .... 15, 16, -17, -18, 19, 20, -21, and so on. UUDDUUD.

Notice how all the numbers in the sequence repeat themselves two or three times.

Only 3/7th of the integers appear this sequence, namely 7 times the triangular numbers plus (0,4,5) or (5,4,0). The numbers can be placed in a grid as follows.

Grid[Transpose[Table[7 n (n - 1)/2 + If[OddQ[n], {0, 4, 5}, {5, 4, 0}], {n, 1, 10}]]]  

0 . 12 . 21 . 47 . 70 . 110 . 147 . 201 . 252 . 320
4 . 11 . 25 . 46 . 74 . 109 . 151 . 200 . 256 . 319
5 . 07 . 26 . 42 . 75 . 105 . 152 . 196 . 257 . 315

King tour through the integer differences

Try it yourself -- start with 4, 5, 7, 4, 0, 5 -- that last move wouldn't be a king move, but from then on all moves are king moves. From the 5, find a move that will increase or decrease 6, then increase or decrease 7, then increase or decrease 8, ...

POSTED BY: Ed Pegg
6 Replies

I've added the Demonstration Semigraceful Eulerian Graphs for exploring these objects.

enter image description here

POSTED BY: Ed Pegg

Nice Demonstration, Ed !

POSTED BY: Vitaliy Kaurov

The Scotty Dog Graph

Graph[#[[1]] \[DirectedEdge] #[[2]] & /@ 
Partition[FoldList[Plus, 0,  Flatten[Table[2 IntegerDigits[73, 2] - 1, {9}]] Range[63]], 2, 1]]

Scotty Dog Graph

Here are some of the more interesting weave generation numbers:

weaving numbers

POSTED BY: Ed Pegg

I think the Uud's were on Doctor Who (maybe spelled Ood).

Ed, I find this unusual. Is there a reason for this sequence?

Here is a version with a single graphic (so it is easier to annotate with arrows):

texts = MapIndexed[Text, 
  Table[7 n (n - 1)/2 + If[OddQ[n], {0, 4, 5}, {5, 4, 0}], {n, 1, 
    10}], {2}];
FindNextNumbers[Text[n_, {x_, y_}], texts_, dx_] := 
 Select[texts, Abs[#[[1]] - n] == dx &]

list = Module[{dx = 5}, 
  Most[NestWhileList[
     With[{choices = FindNextNumbers[#, Flatten[texts], ++dx]},
       Switch[Length[choices], 0, None,
        1, choices[[1]],
        _, Print["error ",Length[choices]]; None]] &, 
     texts[[1, 3]], # =!= None &]][[All, 2]]];

Graphics[{texts, {Red, Arrow /@ Partition[list, 2, 1]}}]

enter image description here

Of course, I wonder about other chess move sequences.

POSTED BY: Todd Rowland

Ed, is UUDDUUD is some known thing, what does it stand for? Also what is a "king move"? Is this post only for mathematicians on an average person can understand this?

POSTED BY: Sam Carrettie

The start is 4, 5, 7, 4, 0, 5, 11, 4 ... up 1, up 2, down 3, down 4, up 5, up 6, down 7. +1, +2, -3, -4, +5, +6, -7. ++--++- ... UUDDUUD.

In Chess, a King moves one space in any direction.

To understand the grid below, all that is needed is to understand subtraction and moving a space.

King grid

POSTED BY: Ed Pegg
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