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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

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

CROSSREFS

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

Sequence in context: A049066 A151688 A159276 * A115947 A247653 A061228

Adjacent sequences:  A056939 A056940 A056941 * A056943 A056944 A056945

KEYWORD

easy,nice,nonn

AUTHOR

Henry Bottomley, Jul 13 2000

STATUS

approved

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Last modified February 17 10:59 EST 2019. Contains 320219 sequences. (Running on oeis4.)