A304584 A linear mapping a(n) = x + d*n of pairs of nonnegative integers (x,d), where the pairs are enumerated by antidiagonals.

0, 1, 2, 2, 5, 10, 3, 9, 17, 27, 4, 14, 26, 40, 56, 5, 20, 37, 56, 77, 100, 6, 27, 50, 75, 102, 131, 162, 7, 35, 65, 97, 131, 167, 205, 245, 8, 44, 82, 122, 164, 208, 254, 302, 352, 9, 54, 101, 150, 201, 254, 309, 366, 425, 486, 10, 65, 122, 181, 242, 305, 370, 437, 506, 577, 650, 11

Offset

0,3

Comments

The sequence solves the following riddle, which has been communicated by Klaus Nagel: A flea starts to jump on the nonnegative integers at time = 0 at an unknown location x >= 0 making jumps of unknown, but constant distance d >= 0 at every subsequent time step. By which strategy can the flea be captured with 100% certainty in a finite number of trials? The solution is to hit a(n) at time = n. This works for all enumerations of pairs (x,d) of integers, because eventually any combination of starting location x and jump width d will be addressed.

Links

Rainer Rosenthal, Table of n, a(n) for n = 0..10000

Example

  d:
  5 |  20
  4 |  14  19
  3 |   9  13  18
  2 |   5   8  12  17
  1 |   2   4   7  11  16
  0 |   0   1   3   6  10  15
    |________________________
  x:    0   1   2   3   4   5
.
a(13) = 1 + 13*3 = 40 because the 13th position in the enumeration corresponds to x=1 and d=3.

Maple

pos2pair:=proc(n) local w, k, e; w:=floor(sqrt(2*n)); if w*(w+1)>2*n then k:=w-1; else k:=w; fi; e:=n-k*(k+1)/2; return [k-e, e]; end:WhereFlea:=proc(n) local x, d, pair; pair:=pos2pair(n); x:=pair[1]; d:=pair[2]; return x+d*n; end:
seq(WhereFlea(n), n=0..66); # Rainer Rosenthal, May 23 2018

Crossrefs

Cf. A304585, A304586, A304587, A305260.

Sequence in context: A184713 A110182 A309867 * A193899 A334017 A208567

Adjacent sequences: A304581 A304582 A304583 * A304585 A304586 A304587

Keyword

nonn

Author

Hugo Pfoertner, May 15 2018

Status

approved