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

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

Author

Henry Bottomley, Jul 13 2000

Keywords

• easy
• nice
• nonn

Comments

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

Examples

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

Links

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

Crossrefs

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

Programs

Haskell

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

Mathematica

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

Python

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

Formula

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).