A056943 Unused area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.

0, 0, 0, 1, 2, 1, 3, 5, 4, 3, 6, 9, 8, 7, 6, 10, 14, 13, 12, 11, 10, 15, 20, 19, 18, 17, 16, 15, 21, 27, 26, 25, 24, 23, 22, 21, 28, 35, 34, 33, 32, 31, 30, 29, 28, 36, 44, 43, 42, 41, 40, 39, 38, 37, 36, 45, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 55, 65, 64, 63, 62, 61, 60, 59

Offset: 0

Author

Henry Bottomley, Jul 13 2000

Keywords

• easy
• nonn

Examples

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

Crossrefs

Cf. A002024, A003056, A056942, A056944.

Programs

Mathematica

uar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2},Floor[c]Ceiling[c]-n]; Array[ uar,80,0] (* Harvey P. Dale, Oct 29 2015 *)

Python

from math import isqrt
def A056943(n): return (isqrt(n<<3)+1>>1)*((k:=isqrt(m:=n+1<<1))-((m>=k*(k+1)+1)^1))-n # Chai Wah Wu, Jun 10 2025

Formula

a(n) =floor[(sqrt(8n+1)-1)/2]*ceiling[(sqrt(8n+1)-1)/2]-n =A002024(n)*A003056(n)-n =A056942(n)-n =n-A056944(n). If n=t(t+1)/2 then a(n)=t(t-1)/2; if n=t(t+1)/2+k with 0

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