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%I #22 Nov 09 2018 18:27:08
%S 0,1,2,-1,-4,2,7,14,7,-2,-11,-22,-11,3,16,31,48,33,16,-3,-22,-43,-66,
%T -45,-22,4,29,56,85,116,89,60,29,-4,-37,-72,-109,-148,-113,-76,-37,5,
%U 46,89,134,181,230,187,142,95,46,-5,-56,-109,-164,-221,-280,-227,-172,-115,-56,6,67
%N A linear mapping a(n) = x + d*n of pairs of integers (x,d), where the pairs are enumerated by a number spiral along antidiagonals.
%C The sequence is an alternative 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="/A304587/b304587.txt">Table of n, a(n) for n = 0..10000</a>
%e d:
%e 3 | 16 28
%e | / \ \
%e 2 | 17 7 15 27
%e | / / \ \ \
%e 1 | 18 8 2 6 14 26
%e | / / / \ \ \ \
%e 0 | 19 9 3 0---1 5 13 25
%e | \ \ \ --> --> -->
%e -1 | 20 10 4 12 24
%e | \ \ / /
%e -2 | 21 11 23
%e | \ /
%e -3 | 22
%e __________________________________
%e x: -3 -2 -1 0 1 2 3 4
%e .
%e a(10) = -1 + 10*(-1) = -11 because the 10th position in the spiral corresponds to x = -1 and d = -1,
%e a(15) = 1 + 15*2 = 31 because the 15th position in the spiral corresponds to x = 1 and d = 2,
%e a(25) = 4 + 25*0 = 4 because the 25th position in the spiral corresponds to x = 4 and d = 0.
%p n2left := proc(n)local w,k;return floor(sqrt((n-1)/2));end:pos2pH:=proc(n)local k,q,Q,e,E,sp;k:=n2left(n);q:=2*k^2+1;Q:=2*(k+1)^2+1;e:=n-q;E:=Q-n;if n<2 then return[n,0];fi;if e<=k then return[-k+e,-e];elif e<2*k then return[-k+e,-2*k+e];elif E<=k+1 then return[-(k+1)+E,E];else return[E-(k+1),2*(k+1)-E];fi;end:WhereFlea:=proc(n) local x,d,pair; pair:=pos2pH(n);x:=pair[1];d:=pair[2];return x+d*n;end: seq(WhereFlea(n),n=0..62);# _Rainer Rosenthal_, May 28 2018
%o (Sage)
%o def a(n):
%o if n<2: return n
%o k = isqrt((n-1)/2)
%o e = n-k*(2*k+1)-1
%o x = e if e<k else k*(2*k+3)-n+2
%o d = abs(x)-k if e<k else k+1-abs(x)
%o return x + d*n
%o print([a(n) for n in range(63)]) # _Peter Luschny_, May 29 2018
%Y Cf. A001844 (where the spiral jumps to next ring), A304584, A304585, A304586.
%K sign,look
%O 0,3
%A _Hugo Pfoertner_, May 16 2018
%E a(1) corrected by _Rainer Rosenthal_, May 28 2018
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