A304586 - OEIS

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.

%I #26 Oct 21 2019 02:23:36

%S 0,1,3,3,3,-1,-7,-7,-7,-7,2,13,26,27,28,29,30,15,-2,-21,-42,-43,-44,

%T -45,-46,-47,-23,3,31,61,93,95,97,99,101,103,105,71,35,-3,-43,-85,

%U -129,-131,-133,-135,-137,-139,-141,-143,-96,-47,4,57,112,169,228,231,234,237,240,243

%N 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.

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

%H Rainer Rosenthal, <a href="/A304586/b304586.txt">Table of n, a(n) for n = 0..10000</a>

%e This is the standard counterclockwise square spiral starting at 0. - _N. J. A. Sloane_, Oct 17 2019

%e d:

%e 3 | 36--35--34--33--32--31--30 55

%e | | | |

%e 2 | 37 16--15--14--13--12 29 54

%e | | | | | |

%e 1 | 38 17 4---3---2 11 28 53

%e | | | | | | | |

%e 0 | 39 18 5 0---1 10 27 52

%e | | | | | | |

%e -1 | 40 19 6---7---8---9 26 51

%e | | | | |

%e -2 | 41 20--21--22--23--24--25 50

%e | | |

%e -3 | 42--43--44--45--46--47--48--49

%e _________________________________

%e x: -3 -2 -1 0 1 2 3 4

%e .

%e a(9) = 2 + 9*(-1) = -7 because the 9th position in the spiral corresponds to x = 2 and d = -1,

%e a(14) = 0 + 14*2 = 28 because the 14th position in the spiral corresponds to x = 0 and d = 2,

%e a(25) = 3 + 25*(-2) = -47 because the 25th position in the spiral corresponds to x = 3 and d = -2.

%p 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

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

%K sign,look

%O 0,3

%A _Hugo Pfoertner_, May 16 2018

%E a(1) and a(2) corrected by _Rainer Rosenthal_, May 24 2018