A304585 -id:A304585 - cp's OEIS frontend

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

Hugo Pfoertner, May 16 2018

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

			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.

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

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

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 eRainer Rosenthal, May 24 2018

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

A304584 A linear mapping a(n) = x + d*n of pairs of nonnegative integers (x,d), where the pairs are enumerated by antidiagonals.

Original entry on oeis.org

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

Offset: 0

Hugo Pfoertner, May 15 2018

nonn

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.

			  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.

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

Cf. A304585, A304586, A304587, A305260.

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

A304587 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.

Original entry on oeis.org

0, 1, 2, -1, -4, 2, 7, 14, 7, -2, -11, -22, -11, 3, 16, 31, 48, 33, 16, -3, -22, -43, -66, -45, -22, 4, 29, 56, 85, 116, 89, 60, 29, -4, -37, -72, -109, -148, -113, -76, -37, 5, 46, 89, 134, 181, 230, 187, 142, 95, 46, -5, -56, -109, -164, -221, -280, -227, -172, -115, -56, 6, 67

Offset: 0

Hugo Pfoertner, May 16 2018

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.

			   d:
   3 |              16  28
     |             /   \   \
   2 |          17   7  15  27
     |         /   /   \   \   \
   1 |      18   8   2   6  14  26
     |     /   /   /   \   \   \   \
   0 |  19   9   3   0---1   5  13  25
     |     \   \   \    --> --> -->
  -1 |      20  10   4  12  24
     |         \   \  /   /
  -2 |          21  11  23
     |             \   /
  -3 |              22
    __________________________________
  x:    -3  -2  -1   0   1   2   3   4
.
a(10) = -1 + 10*(-1) = -11 because the 10th position in the spiral corresponds to x = -1 and d = -1,
a(15) = 1 + 15*2 = 31 because the 15th position in the spiral corresponds to x = 1 and d = 2,
a(25) = 4 + 25*0 = 4 because the 25th position in the spiral corresponds to x = 4 and d = 0.

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

Cf. A001844 (where the spiral jumps to next ring), A304584, A304585, A304586.

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

def a(n):
    if n<2: return n
    k = isqrt((n-1)/2)
    e = n-k*(2*k+1)-1
    x = e if ePeter Luschny, May 29 2018

a(1) corrected by Rainer Rosenthal, May 28 2018

A305260 A linear mapping a(n) = x + y*n of pairs of nonnegative integers (x,y), where the pairs are enumerated first by radial coordinate r and in case of ties, by polar angle 0 <= phi <= Pi/2 in a polar coordinate system.

Original entry on oeis.org

0, 1, 2, 4, 2, 10, 8, 15, 18, 3, 30, 14, 37, 29, 44, 4, 64, 21, 73, 60, 44, 86, 5, 73, 99, 125, 31, 136, 61, 147, 124, 98, 163, 6, 204, 41, 217, 80, 230, 161, 204, 129, 255, 7, 308, 52, 235, 330, 198, 298, 107, 359, 163, 374, 276, 335, 8, 456, 66, 243, 424, 489, 132, 506, 390, 203, 531

Offset: 0

Hugo Pfoertner, Jun 15 2018

nonn

Secondary sorting by polar angle is equivalent to secondary sorting by y.

The sequence is an alternative solution to the riddle described in the comments of A304584.

			   y:
     |
   8 |  57  61  63  66  70
     |
   7 |  44  47  51  53  60  68
     |
   6 |  34  36  38  42  49  55  64
     |
   5 |  25  27  29  32  40  46  54  67
     |
   4 |  16  18  21  24  30  39  48  59  69
     |
   3 |  10  12  14  19  23  31  41  52  65
     |
   2 |   5   7   8  13  20  28  37  50  62
     |
   1 |   2   3   6  11  17  26  35  45  58
     |
   0 |   0   1   4   9  15  22  33  43  56  71
       _______________________________________
  x:     0   1   2   3   4   5   6   7   8   9
.
a(5) = x(5) + 5*y(5) = 0 + 5*2 = 10,
a(14) = x(14) + 14*y(14) = 2 + 14*3 = 44,
a(20) = x(20) + 20*y(20) = 4 + 20*2 = 44.

Cf. A000925, A283305, A283306, A304584, A304585.

n=-1;for(r2=0,81,for(y=0,round(sqrt(r2)),x2=r2-y^2;if(issquare(x2),print1(round(sqrt(x2))+y*(n++),", "))))

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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.

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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.

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A304587 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.

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A305260 A linear mapping a(n) = x + y*n of pairs of nonnegative integers (x,y), where the pairs are enumerated first by radial coordinate r and in case of ties, by polar angle 0 <= phi <= Pi/2 in a polar coordinate system.

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