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

0,3

Comments

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

Links

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

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).

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A151688 A159276 A359362 * A115947 A247653 A061228

Adjacent sequences: A056939 A056940 A056941 * A056943 A056944 A056945

Keyword

easy,nice,nonn

Author

Henry Bottomley, Jul 13 2000

Status

approved