A232115 -id:A232115 - cp's OEIS frontend

A232113 a(n) is the Manhattan distance between n and 2*n in a square spiral of positive integers with 1 at the center.

1, 2, 3, 2, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 7, 8, 9, 10, 11, 8, 11, 12, 13, 12, 9, 10, 13, 16, 15, 12, 9, 12, 15, 16, 15, 12, 9, 12, 15, 14, 13, 12, 11, 12, 13, 12, 11, 10, 9, 10, 11, 10, 9, 8, 7, 8, 9

Offset: 1

Alex Ratushnyak, Nov 19 2013

nonn

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: 1, 4, 16, 625, 20736, 707281, 24010000, 815730721, ...; that is, 1, 4, 2^4, 5^4, 12^4, 29^4, 70^4, 169^4, ... (cf. Pell numbers: A000129).

Cf. A000129, A214526, A232114, A232115.

A232114 a(n) is the Manhattan distance between n and n^2 in a square spiral of positive integers with 1 at the center.

Original entry on oeis.org

0, 2, 2, 2, 6, 4, 6, 8, 6, 12, 8, 12, 12, 12, 16, 12, 20, 14, 20, 18, 20, 22, 20, 26, 20, 30, 22, 30, 26, 30, 30, 30, 34, 30, 38, 30, 42, 32, 42, 36, 42, 40, 42, 44, 42, 48, 42, 52, 42, 56, 44, 56, 48, 56, 52, 56, 56, 56, 60, 56, 64, 56, 68, 56, 72, 58, 72, 62, 72, 66

Offset: 1

Alex Ratushnyak, Nov 19 2013

nonn

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)=n: 2, 8, 12, 22, 30, 44, 56, 74, 90, 112, 132, 158, 182, 212, 240, 274, 306, 344, 380, 422, ... That is, 2 * A084263.

Cf. A000290, A084263, A214526, A232113, A232115.

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.

Original entry on oeis.org

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

Alex Ratushnyak, Jan 16 2014

nonn

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

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

Cf. A214526, A232113, A232114, A232115.

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=', ')

cp's OEIS Frontend

A232113 a(n) is the Manhattan distance between n and 2*n in a square spiral of positive integers with 1 at the center.

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A232114 a(n) is the Manhattan distance between n and n^2 in a square spiral of positive integers with 1 at the center.

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Author

Keywords

Comments

Crossrefs

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.

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Python