A305260 -id:A305260 - cp's OEIS frontend

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

Hugo Pfoertner, May 15 2018

nonn

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.

			  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.

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

Cf. A304585, A304586, A304587, A305260.

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

cp's OEIS Frontend

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

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