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

Author

Alex Ratushnyak, Nov 19 2013

Keywords

• nonn

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, ...

Crossrefs

Cf. A000217, A214526, A232113, A232114.

Programs

Python

import math
def get_x_y(n):
  sr = math.isqrt(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(abs(x1-x0)+abs(y1-y0), end=', ')