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A304584 A linear mapping a(n) = x + d*n of pairs of nonnegative integers (x,d), where the pairs are enumerated by antidiagonals. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A003228 A184713 A110182 * A193899 A208567 A273968

Adjacent sequences:  A304581 A304582 A304583 * A304585 A304586 A304587

KEYWORD

nonn

AUTHOR

Hugo Pfoertner, May 15 2018

STATUS

approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)