A235913 a(n) is the Manhattan distance between n^3 and (n+1)^3 in a square spiral of positive integers with 1 at the center.

1, 3, 11, 15, 13, 9, 5, 21, 33, 59, 71, 49, 47, 35, 15, 13, 43, 73, 109, 123, 117, 109, 167, 141, 113, 77, 43, 5, 51, 95, 145, 201, 263, 281, 397, 413, 317, 333, 269, 239, 183, 121, 63, 11, 81, 147, 219, 307, 379, 471, 567, 623, 517, 569, 683, 503, 545, 473, 395, 311

Offset

1,2

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

Links

Table of n, a(n) for n=1..60.

Example

Manhattan distance between 2^3=8 and 3^3=27 is 3 in a square spiral, so a(2)=3.

Prog

(Python)
import math
def get_x_y(n):
  sr = int(math.sqrt(n-1))  # Ok for small n's
  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, 77):
  x0, y0 = get_x_y(n**3)
  x1, y1 = get_x_y((n+1)**3)
  print(abs(x1-x0)+abs(y1-y0), end=', ')

Crossrefs

Cf. A214526, A232113, A232114, A232115.

Sequence in context: A273869 A289049 A288980 * A132363 A134629 A245522

Adjacent sequences: A235910 A235911 A235912 * A235914 A235915 A235916

Keyword

nonn

Author

Alex Ratushnyak, Jan 16 2014

Status

approved