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.

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

Author

Alex Ratushnyak, Jan 23 2014

Keywords

• nonn

Examples

			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
		

Links

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

Crossrefs

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

Programs

Python

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=', ')

Formula

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