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

1,3

Links

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

Formula

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

Example

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

Prog

(Python)
import math
def getXY(n):
  y = math.isqrt(n*2)
  if n<=y*(y+1)//2: y-=1
  x = n - y*(y+1)//2
  return x, y
def a(n):
  ox, oy = getXY(n)
  nx, ny = getXY(2*n)
  return abs(nx-ox)+abs(ny-oy)
print([a(n) for n in range(1, 83)])

Crossrefs

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

Sequence in context: A237817 A306949 A189025 * A045430 A067693 A322006

Adjacent sequences: A236344 A236345 A236346 * A236348 A236349 A236350

Keyword

nonn,easy

Author

Alex Ratushnyak, Jan 23 2014

Status

approved