|
|
A056942
|
|
Area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.
|
|
3
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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..]
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nice,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|