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
Examples
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
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2 | 37 16--15--14--13--12 29 54
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1 | 38 17 4---3---2 11 28 53
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0 | 39 18 5 0---1 10 27 52
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-1 | 40 19 6---7---8---9 26 51
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-2 | 41 20--21--22--23--24--25 50
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-3 | 42--43--44--45--46--47--48--49
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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.
Links
- Rainer Rosenthal, Table of n, a(n) for n = 0..10000
Programs
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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
Rainer Rosenthal, May 24 2018
Extensions
a(1) and a(2) corrected by Rainer Rosenthal, May 24 2018
Comments