A232113 -id:A232113 - cp's OEIS frontend

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

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.

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.

Original entry on oeis.org

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

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: 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, ...

Cf. A000217, A214526, A232113, A232114.

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=', ')

A236347 Manhattan distances between n and 2*n in a left-aligned triangle with next M natural numbers in row M: 1, 2 3, 4 5 6, 7 8 9 10, etc.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 2, 3, 2, 3, 4, 5, 6, 5, 6, 7, 5, 4, 3, 9, 4, 3, 4, 5, 6, 10, 5, 6, 7, 8, 9, 8, 7, 6, 10, 11, 12, 6, 5, 4, 5, 6, 7, 4, 5, 6, 7, 8, 9, 10, 12, 11, 10, 10, 11, 12, 13, 14, 9, 8, 7, 6, 5, 6, 17, 18, 6, 5, 6, 7, 8, 9, 10, 11, 16, 15, 9, 10, 11, 12

Offset: 1

Alex Ratushnyak, Jan 23 2014

nonn

			Triangle in which we find distances begins:
_1
_2  3
_4  5  6
_7  8  9 10
11 12 13 14 15
16 17 18 19 20 21
22 23 24 25 26 27 28
29 30 31 32 33 34 35 36
37 38 39 40 41 42 43 44 45

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A002260, A002024, A232113, A236345, A236346.

import math
def getXY(n):
  y = int(math.sqrt(n*2))
  if n<=y*(y+1)//2: y-=1
  x = n - y*(y+1)//2
  return x, y
for n in range(1,88):
  ox, oy = getXY(n)
  nx, ny = getXY(2*n)
  print(abs(nx-ox)+abs(ny-oy), end=', ')

a(n) = abs(A002260(2*n) - A002260(n)) + A002024(2*n) - A002024(n). - David Radcliffe, Aug 06 2025

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

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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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.

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Python

A236347 Manhattan distances between n and 2*n in a left-aligned triangle with next M natural numbers in row M: 1, 2 3, 4 5 6, 7 8 9 10, etc.

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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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python