A056944 Amount by which used area of rectangle needed to enclose a non-touching spiral of length n on a square lattice exceeds unused area.

0, 1, 2, 2, 2, 4, 3, 2, 4, 6, 4, 2, 4, 6, 8, 5, 2, 4, 6, 8, 10, 6, 2, 4, 6, 8, 10, 12, 7, 2, 4, 6, 8, 10, 12, 14, 8, 2, 4, 6, 8, 10, 12, 14, 16, 9, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 12, 2, 4, 6, 8, 10, 12, 14, 16

Offset: 0

Author

Henry Bottomley, Jul 13 2000

Keywords

• easy
• nonn
• nice

Comments

m (when n is m-th triangular number) followed by m even numbers from 2 through 2m.

Examples

			a(9)=6 since spiral is as marked by 9 X's in 4*3 = 12 rectangle, with 12-9 = 3 spaces unused and a used-unused difference of 9-3 = 6:
X.XX
X..X
XXXX
As a triangle, the first few rows are: 1; 2, 2; 2, 4, 3; 2, 4, 6, 4; 2, 4, 6, 8, 5; 2, 4, 6, 8, 10, 6; 2, 4, 6, 8, 10, 12, 7; ... (= reversal of triangle A143595). Row sums = n^2. - _Gary W. Adamson_, Aug 26 2008
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A002024, A003056, A056942, A056943.

Cf. A143595. - Gary W. Adamson, Aug 26 2008

Programs

Magma

[2*n-Floor((Sqrt(8*n+1)-1)/2)*Ceiling((Sqrt(8*n+1)-1)/2): n in [0..90]]; // Vincenzo Librandi, Aug 06 2017

Mathematica

uar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2},2n-Floor[c]Ceiling[c]]; Array[uar,90,0] (* Harvey P. Dale, Aug 14 2013 *)

Python

from math import isqrt
def A056944(n): return (n<<1)-(isqrt(n<<3)+1>>1)*((k:=isqrt(m:=n+1<<1))-((m>=k*(k+1)+1)^1)) # Chai Wah Wu, Jun 10 2025

Formula

a(n) = 2n - floor((sqrt(8n+1)-1)/2)*ceiling((sqrt(8n+1)-1)/2) = 2n - A002024(n)*A003056(n) = 2n - A056942(n) = n -A056943(n). If n = t(t+1)/2 then a(n)=t; if n = t(t+1)/2+k with 0 < k <= t then a(n)=2k.