A214848 -id:A214848 - cp's OEIS frontend

A022846 Nearest integer to n*sqrt(2).

0, 1, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95, 96

Offset: 0

Clark Kimberling

Let R(i,j) be the rectangle with antidiagonals 1; 2,3; 4,5,6; ...; n^2 is in antidiagonal number a(n). Proof: n^2 is in antidiagonal m iff A000217(m-1)< n^2 <=A000217(m), where A000217(m)=m*(m+1)/2. So m = A002024(n^2) = round(n*sqrt(2)) = a(n). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Mar 07 2003
In the rectangle R(i,j), n^2 is the number in row i=A057049(n) and column j=A057050(n), so that for n >= 1, a(n) = -1 + A057049(n) + A057050(n). - Clark Kimberling, Jan 31 2011
Number of triangular numbers less than n^2. - Philippe Deléham, Mar 08 2013

			n = 4, n^2 = 16; 0, 1, 3, 6, 10, 15 are triangular numbers in interval [0, 16); a(4) = 6. - _Philippe Deléham_, Mar 08 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.

Cf. A063957 (complement of this set).

Cf. A214848 (first differences), also A006338.

a022846 = round . (* sqrt 2) . fromIntegral
-- Reinhard Zumkeller, Mar 03 2014

[Round(n*Sqrt(2)): n in [0..60]]; // Vincenzo Librandi, Oct 22 2011

Round[Sqrt[2]Range[0,70]] (* Harvey P. Dale, Jun 18 2013 *)

PARI
```
a(n)=round(n*sqrt(2))
```

from math import isqrt
def A022846(n): return isqrt(n**2<<3)+1>>1 # Chai Wah Wu, Feb 10 2023

a(n) = A002024(n^2).

a(n+1) - a(n) = 1 or 2. - Philippe Deléham, Mar 08 2013

A006338 An "eta-sequence": floor((n+1)sqrt(2) + 1/2) - floor(nsqrt(2) + 1/2).

Original entry on oeis.org

2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2

Offset: 1

D. R. Hofstadter, Jul 15 1977

Equals its own "second derivative" (cf. A006337).
Presumably this is the same as the following sequence from Hofstadter's book: the number of triangular numbers between each successive pair of squares. More precisely, a(n) is the number of triangular numbers T such that n^2 <= T < (n+1)^2. E.g., a(3) = 2 because 3^2 <= T < 4^2 permits T(4) = 10 and T(5) = 15 and no other triangular number. - Hugo van der Sanden, May 03 2005.
a(n) = A214848(n) = A022846(n+1) - A022846(n). - Reinhard Zumkeller, Mar 03 2014

Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a sequence".
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991

Cf. A006337, A022846, A214848.

a006338 n = a006338_list !! (n-1)
a006338_list = tail a214848_list
-- Reinhard Zumkeller, Mar 03 2014

[Floor((n+1)*Sqrt(2)+1/2) - Floor(n*Sqrt(2)+1/2): n in [1..30]]; // G. C. Greubel, Nov 18 2017

a[n_] := Floor[(n+1)*Sqrt[2]+1/2] - Floor[n*Sqrt[2]+1/2]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 24 2015 *)
Differences[Table[Floor[n Sqrt[2]+1/2],{n,120}]] (* Harvey P. Dale, Dec 10 2021 *)

for(n=1,30, print1(floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2), ", ")) \\ G. C. Greubel, Nov 18 2017

a(n) = floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2). - G. C. Greubel, Nov 18 2017

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 28 2003

A214857 Number of triangular numbers in interval [0, n^2].

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 34, 35, 37, 38, 40, 41, 42, 44, 45, 47, 48, 50, 51, 52, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 68, 69, 71, 72, 74, 75, 76, 78, 79, 81, 82, 83, 85, 86, 88, 89, 91, 92, 93, 95, 96, 98, 99, 100

Offset: 0

Philippe Deléham, Mar 09 2013

Partial sums of A214856.

			0, 1, 3, 6 are in interval [0, 9], a(3) = 4.
0, 1, 3, 6, 10, 15 are in interval [0, 16], a(4) = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A022846, A214848, A214856.

nn = 100; tri = Table[n (n + 1)/2, {n, 0, nn}]; Table[Count[tri, ?(# <= n^2 &)], {n, 0, Sqrt[tri[[-1]]]}] (* _T. D. Noe, Mar 11 2013 *)
Table[Floor[(Sqrt[8*n^2+1]-1)/2]+1,{n,0,80}] (* Harvey P. Dale, Oct 14 2014 *)

from math import isqrt
def A214857(n): return isqrt(n**2+1<<3)+1>>1 # Chai Wah Wu, Dec 09 2024

a(n) = floor((1 + sqrt(1+8*n^2))/2). - Ralf Stephan, Jan 30 2014

A214856 Number of triangular numbers in interval ](n-1)^2, n^2] for n>0, a(0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1

Offset: 0

Philippe Deléham, Mar 08 2013

			10, 15 are in interval ]9, 16] , a(4) = 2.

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A214848, A214857.

a(n) = if (n, sum(i=(n-1)^2+1, n^2, ispolygonal(i, 3)), 1); \\ Michel Marcus, Nov 12 2022

from math import isqrt
def A214856(n): return (isqrt((m:=n**2<<3)+8)+1>>1)-(isqrt(m-(n-1<<4))+1>>1) if n else 1 # Chai Wah Wu, Dec 09 2024

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A022846 Nearest integer to n*sqrt(2).

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A006338 An "eta-sequence": floor((n+1)*sqrt(2) + 1/2) - floor(n*sqrt(2) + 1/2).

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A214857 Number of triangular numbers in interval [0, n^2].

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A214856 Number of triangular numbers in interval ](n-1)^2, n^2] for n>0, a(0)=1.

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PARI

Python

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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A006338 An "eta-sequence": floor((n+1)sqrt(2) + 1/2) - floor(nsqrt(2) + 1/2).