A304586 A linear mapping a(n) = x + d*n of pairs of integers (x,d), where the pairs are enumerated by the counterclockwise square spiral (an axis-parallel number spiral) starting at 0.

0, 1, 3, 3, 3, -1, -7, -7, -7, -7, 2, 13, 26, 27, 28, 29, 30, 15, -2, -21, -42, -43, -44, -45, -46, -47, -23, 3, 31, 61, 93, 95, 97, 99, 101, 103, 105, 71, 35, -3, -43, -85, -129, -131, -133, -135, -137, -139, -141, -143, -96, -47, 4, 57, 112, 169, 228, 231, 234, 237, 240, 243

Offset

0,3

Comments

The sequence is a solution to the riddle described in the comments of A304584 without the restriction of x and d to nonnegative numbers.

Links

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

Example

This is the standard counterclockwise square spiral starting at 0. - N. J. A. Sloane, Oct 17 2019
d:
   3 |  36--35--34--33--32--31--30  55
     |   |                       |   |
   2 |  37  16--15--14--13--12  29  54
     |   |   |               |   |   |
   1 |  38  17   4---3---2  11  28  53
     |   |   |   |       |   |   |   |
   0 |  39  18   5   0---1  10  27  52
     |   |   |   |           |   |   |
  -1 |  40  19   6---7---8---9  26  51
     |   |   |                   |   |
  -2 |  41  20--21--22--23--24--25  50
     |   |                           |
  -3 |  42--43--44--45--46--47--48--49
     _________________________________
  x:    -3  -2  -1   0   1   2   3   4
.
a(9) = 2 + 9*(-1) = -7 because the 9th position in the spiral corresponds to x = 2 and d = -1,
a(14) = 0 + 14*2 = 28 because the 14th position in the spiral corresponds to x = 0 and d = 2,
a(25) = 3 + 25*(-2) = -47 because the 25th position in the spiral corresponds to x = 3 and d = -2.

Maple

square2pair:=proc(sq)local w, k; w:=floor(sqrt(sq)); k:=floor(w/2); if modp(sq, 2)=0 then return[-k, k]; else return[k+1, -k]; fi; end:pos2pS:=proc(n)local w, q, Q, e, E, sp; w:=floor(sqrt(n)); q := w^2; Q:=(w+1)^2; e:=n-q; E:=Q-n; if e<E then sp:=square2pair(q); if modp(q, 2)=0 then return[sp[1], sp[2]-e]; else return[sp[1], sp[2]+e]; fi; else sp:=square2pair(Q); if modp(Q, 2)=0 then return[sp[1]+E, sp[2]]; else return[sp[1]-E, sp[2]]; fi; fi; end:WhereFlea:=proc(n) local x, d, pair; pair:=pos2pS(n); x:=pair[1]; d:=pair[2]; return x+d*n; end: seq(WhereFlea(n), n=0..61); # Rainer Rosenthal, May 24 2018

Crossrefs

Cf. A174344, A274923, A304584, A304585, A304587.

Sequence in context: A014465 A226645 A243095 * A155969 A076237 A201432

Adjacent sequences: A304583 A304584 A304585 * A304587 A304588 A304589

Keyword

sign,look

Author

Hugo Pfoertner, May 16 2018

Extensions

a(1) and a(2) corrected by Rainer Rosenthal, May 24 2018

Status

approved