A232115 a(n) is the Manhattan distance between n and n(n+1)/2 in a square spiral of positive integers with 1 at the center.

0, 1, 3, 4, 2, 3, 5, 6, 4, 5, 9, 6, 6, 7, 11, 8, 8, 15, 9, 10, 10, 13, 11, 12, 22, 11, 15, 18, 12, 15, 17, 22, 12, 21, 25, 10, 26, 21, 19, 18, 24, 27, 15, 34, 24, 21, 31, 20, 28, 21, 31, 24, 28, 41, 19, 36, 36, 23, 35, 26, 38, 23, 41, 36, 28, 53, 29, 38, 40, 31, 39

Offset

1,3

Comments

Spiral begins:

.

49 26--27--28--29--30--31

| | |

48 25 10--11--12--13 32

| | | | |

47 24 9 2---3 14 33

| | | | | | |

46 23 8 1 4 15 34

| | | | | |

45 22 7---6---5 16 35

| | | |

44 21--20--19--18--17 36

| |

43--42--41--40--39--38--37

.

Numbers n such that a(n)*2=n: 2, 6, 10, 12, 14, 16, 20, 24, 30, 80, 192, 198, 350, 524, 536, 548, 552, 560, 564, 576, 588, 594, 606, 618, 630, 1380, 1900, 4446, ...

Links

Table of n, a(n) for n=1..71.

Prog

(Python)
import math
def get_x_y(n):
  sr = int(math.sqrt(n-1))
  sr = sr-1+(sr&1)
  rm = n-sr*sr
  d = (sr+1)/2
  if rm<=sr+1:
     return -d+rm, d
  if rm<=sr*2+2:
     return d, d-(rm-(sr+1))
  if rm<=sr*3+3:
     return d-(rm-(sr*2+2)), -d
  return -d, -d+rm-(sr*3+3)
for n in range(1, 333):
  x0, y0 = get_x_y(n)
  x1, y1 = get_x_y(n*(n+1)/2)
  print str(abs(x1-x0)+abs(y1-y0))+', ',

Crossrefs

Cf. A000217, A214526, A232113, A232114.

Sequence in context: A121891 A346780 A271590 * A275529 A326044 A011414

Adjacent sequences: A232112 A232113 A232114 * A232116 A232117 A232118

Keyword

nonn

Author

Alex Ratushnyak, Nov 19 2013

Status

approved