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

0,5

Links

Table of n, a(n) for n=0..73.

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<k <= t then a(n)=t(t+1)/2-k.

Example

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

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 *)

Crossrefs

Cf. A002024, A003056, A056942, A056944.

Sequence in context: A257161 A253676 A182939 * A258654 A171085 A288538

Adjacent sequences: A056940 A056941 A056942 * A056944 A056945 A056946

Keyword

easy,nonn

Author

Henry Bottomley, Jul 13 2000

Status

approved