A236346 Manhattan distances between n^2 and (n+1)^2 in a left-aligned triangle with next M natural numbers in row M: 1, 2 3, 4 5 6, 7 8 9 10, etc.

2, 3, 4, 4, 5, 6, 6, 8, 7, 10, 8, 9, 12, 10, 14, 11, 12, 16, 13, 18, 14, 20, 15, 16, 22, 17, 24, 18, 19, 26, 20, 28, 21, 22, 30, 23, 32, 24, 34, 25, 26, 36, 27, 38, 28, 29, 40, 30, 42, 31, 44, 32, 33, 46, 34, 48, 35, 36, 50, 37, 52, 38, 54, 39, 40, 56, 41, 58

Offset: 1

Author

Alex Ratushnyak, Jan 23 2014

Keywords

• nonn
• easy

Comments

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

Subsequence of terms such that a(m)>=a(m-1) and a(m)>=a(m+1) seems to be A005843 (even numbers) except first two terms, and if such a(m) are removed, the remainder seems to be A000027 (natural numbers) except 1:

2, 3, *4*, 4, 5, *6*, 6, *8*, 7, *10*, 8, 9, *12*, 10, *14*, 11, 12, *16*, 13, *18*, 14, *20*, 15, ...

Links

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

Crossrefs

Cf. A236345, A005843, A000027.

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, 77):
  ox, oy = getXY(n*n)
  nx, ny = getXY((n+1)**2)
  print(abs(nx-ox)+abs(ny-oy), end=', ')

Formula

a(n) = A214848(n) + |A057049(n+1) - A057049(n)|. - David Radcliffe, Aug 06 2025