login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A226324 Array by antidiagonals:  D(m,n) = distance between m and n using the graph-metric of A226247. 1
0, 1, 1, 2, 0, 2, 2, 1, 1, 2, 3, 1, 0, 1, 3, 3, 2, 2, 2, 2, 3, 4, 2, 1, 0, 1, 2, 4, 4, 3, 1, 3, 3, 1, 3, 4, 4, 3, 2, 3, 0, 3, 2, 3, 4, 5, 3, 2, 4, 2, 2, 4, 2, 3, 5, 5, 4, 2, 4, 1, 0, 1, 4, 2, 4, 5, 5, 4, 3, 4, 1, 3, 3, 1, 4, 3, 4, 5, 5, 4, 3, 5, 3, 3, 0, 3 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Let S be the set of numbers defined by these rules:  0 is in S; if x is in S, then x+1 is in S, and if nonzero x is in S, then -1/x is in S.  Then S is the set of all rational numbers, produced in generations as follows:

g(1) = (0), g(2) = (1), g(3) = (2, -1), g(4) = (3, -1/2), g(5) = (4,-1/3,1/2),... For n > 2, once g(n-1) = (c(1),...,c(z)) is defined, g(n) is formed from the vector (c(1)+1, -1/c(1), c(2)+1, -1/c(2),...,c(z)+1, -1/c(z)) by deleting previously generated elements.  This order of generation matches a tree with (0,1), (1,2), (1,-1), (2,3), (2,-1/2), (3,4), (4,-1/3), (-1/2,1/2), etc.  Replace each node by the order in which it is generated, so that the nodes labeled (0,1,2,-1,3,-1/2,4,-1/3,...) get new labels (1,2,3,4,5,6,...), respectively.  If m and n are positive integers, then D(m,n) is the number of edges between m and n.

LINKS

Clark Kimberling, Antidiagonals n=1..60, flattened

EXAMPLE

Northwest corner of the distance table:

0 1 2 2 3 3 4 4 4 5

1 0 1 1 2 2 3 3 3 4

2 1 0 2 1 1 2 2 2 3

2 1 2 0 3 3 4 4 4 5

3 2 1 3 0 2 1 1 3 2

3 2 1 3 2 0 3 3 1 4

4 3 2 4 1 3 0 2 4 1

4 3 2 4 1 3 2 0 4 3

4 3 2 4 3 1 4 4 0 5

5 4 3 5 2 4 1 3 5 0

Row 5, column 4 is occupied by 3, meaning that D(5,4) = 3, a count of edges in the subgraph 5 -> 3 -> 2 -> 4.

MATHEMATICA

$MaxExtraPrecision = Infinity; g[1] := {1}; g[2] := {1, 0}; g[3] := {1, 0, 0}; g[test_] := Module[{topRow, len, tmp = test, noOfTerms = Ceiling[Log[test]/Log[1.465571231876768026656731]] - 1}, topRow = Flatten[{1, LinearRecurrence[{1, 0, 1}, {2, 3, 5}, noOfTerms]}]; If[First[#] == 0, Rest[#], #] &[Table[If[# >= 0, tmp = #; 1, 0] &[tmp - topRow[[n]]], {n, noOfTerms, 1, -1}]]]; d[n1_, n2_] := Module[{z1 = g[n1], z2 = g[n2]}, Length[z1] + Length[z2] - 2(NestWhile[# + 1 &, 1, z1[[#]] == z2[[#]] &, 1, Min[{Length[z1], Length[z2]}]] - 1)]; (dArray = Table[d[m, n], {m, 1, #}, {n, 1, #}] &[15]) // TableForm

  Flatten[Table[d[k, n + 1 - k], {n, 1, 15}, {k, 1, n}]]

  ArrayPlot[dArray, ColorFunction -> "BlueGreenYellow"]

(* Peter J. C. Moses, Jun 02 2013 *)

CROSSREFS

Cf. A226080, A226207, A226247.

Sequence in context: A129678 A261773 A226207 * A023604 A219660 A060964

Adjacent sequences:  A226321 A226322 A226323 * A226325 A226326 A226327

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Jun 04 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 5 13:26 EST 2019. Contains 329751 sequences. (Running on oeis4.)