A038759 -id:A038759 - cp's OEIS frontend

A038760 a(n) = n - floor(sqrt(n)) * ceiling(sqrt(n)).

0, 0, 0, 1, 0, -1, 0, 1, 2, 0, -2, -1, 0, 1, 2, 3, 0, -3, -2, -1, 0, 1, 2, 3, 4, 0, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 0, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 0, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 0, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, -8, -7, -6, -5, -4

Offset: 0

Henry Bottomley, May 03 2000

			Sqrt(31) is between 5 and 6, and 31 - 6*5 = 1, so a(31)=1.

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

Cf. A053188.

a:= n-> n -(x-> floor(x)*ceil(x))(sqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Jan 03 2015

f[n_]:=n-Floor[Sqrt[n]]*Ceiling[Sqrt[n]];Table[f[n],{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Mar 29 2010 *)

a(n)=if(issquare(n),0,my(s=sqrtint(n));n-s^2-s) \\ Charles R Greathouse IV, Feb 07 2013

from math import isqrt
def A038760(n): return m-k if (m:=n-(k:=isqrt(n))**2) else 0 # Chai Wah Wu, Jul 28 2022

a(n) = n - A000196(n)*A003059(n) = n - A038759(n).

A056942 Area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.

Original entry on oeis.org

0, 1, 2, 4, 6, 6, 9, 12, 12, 12, 16, 20, 20, 20, 20, 25, 30, 30, 30, 30, 30, 36, 42, 42, 42, 42, 42, 42, 49, 56, 56, 56, 56, 56, 56, 56, 64, 72, 72, 72, 72, 72, 72, 72, 72, 81, 90, 90, 90, 90, 90, 90, 90, 90, 90, 100, 110, 110, 110, 110, 110, 110, 110, 110, 110, 110, 121

Offset: 0

Henry Bottomley, Jul 13 2000

m^2 (when n is m-th triangular number) followed by m copies of m-th pronic [m(m+1)].

			a(9) = 12 since spiral is as marked by 9 X's in 4*3 = 12 rectangle:
X.XX
X..X
XXXX

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A000290, A002024, A002378, A002620, A003056, A038759, A056943, A056944.

a056942 n = a056942_list !! n
a056942_list = concatMap
               (\x -> (x ^ 2) : (take x $ repeat (x * (x + 1)))) [0..]
-- Reinhard Zumkeller, Feb 11 2014

ar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2},Floor[c]Ceiling[c]]; Array[ar,70,0] (* Harvey P. Dale, Feb 27 2012 *)

from math import isqrt
def A056942(n): return (isqrt(n<<3)+1>>1)*((k:=isqrt(m:=n+1<<1))-((m>=k*(k+1)+1)^1)) # Chai Wah Wu, Jun 10 2025

a(n) = floor[(sqrt(8n+1)-1)/2]*ceiling[(sqrt(8n+1)-1)/2] = A002024(n)*A003056(n) = A056943(n)+n = 2n-A056944(n). If n = t(t+1)/2 then a(n) = t^2; if t(t-1)/2 < n < t(t+1)/2 then a(n) = t(t-1).

A174803 a(n) = n + ceiling(sqrt(n))*floor(sqrt(n)).

Original entry on oeis.org

0, 2, 4, 5, 8, 11, 12, 13, 14, 18, 22, 23, 24, 25, 26, 27, 32, 37, 38, 39, 40, 41, 42, 43, 44, 50, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 72, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 98, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 128, 137

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 29 2010

Cf. A038759, A038760.

Cf. A000290 (n^2), A001105 (2*n^2).

f[n_]:=n+Floor[Sqrt[n]]*Ceiling[Sqrt[n]];Table[f[n],{n,0,5!}]

a(n) = n + A038759(n). - Robert G. Wilson v, Apr 17 2010

a(n^2) = 2*n^2. - Bernard Schott, Jul 30 2022

cp's OEIS Frontend

A038760 a(n) = n - floor(sqrt(n)) * ceiling(sqrt(n)).

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A056942 Area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.

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A174803 a(n) = n + ceiling(sqrt(n))*floor(sqrt(n)).

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