A236345 -id:A236345 - cp's OEIS frontend

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

Alex Ratushnyak, Jan 23 2014

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, ...

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

Cf. A236345, A005843, A000027.

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=', ')

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

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

A236459 Regular triangle: T(n, k) Manhattan distance between n and k in a left-aligned triangle with next M natural numbers in row M.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 2, 1, 2, 0, 3, 2, 1, 1, 0, 4, 3, 2, 2, 1, 0, 3, 2, 3, 1, 2, 3, 0, 4, 3, 2, 2, 1, 2, 1, 0, 5, 4, 3, 3, 2, 1, 2, 1, 0, 6, 5, 4, 4, 3, 2, 3, 2, 1, 0, 4, 3, 4, 2, 3, 4, 1, 2, 3, 4, 0, 5, 4, 3, 3, 2, 3, 2, 1, 2, 3, 1, 0, 6, 5, 4, 4, 3, 2, 3, 2, 1, 2, 2, 1, 0, 7, 6, 5, 5, 4, 3, 4, 3, 2, 1, 3, 2, 1, 0

Offset: 1

Michel Marcus, Jan 26 2014

First column is A051162. Right diagonal is all zeros.

			Triangle where distances are measured begins:
1
2 3
4 5 6
7 8 9 10
Distance between 1 and 1 is 0, hence T(1, 1) = 0.
Distance between 2 and 1 is 1, and between 2 and 2 is 0. Hence second row of this triangle is 1, 0.
Triangle starts:
0;
1, 0;
2, 1, 0;
2, 1, 2, 0;
3, 2, 1, 1, 0;

Cf. A236345.

getxy(n) = {y = sqrtint(2*n); if (n<=y*(y+1)/2, y--); x = n - y*(y+1)/2; [x, y];}
trg(nn) = {i= 1; for (n = 1, nn, v = getxy(n); for (k = 1, n, nv = getxy(k); print1(abs(nv[1]-v[1])+abs(nv[2]-v[2]), ", ");); print(););}

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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.

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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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A236459 Regular triangle: T(n, k) Manhattan distance between n and k in a left-aligned triangle with next M natural numbers in row M.

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