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Area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.
3

%I #11 Aug 19 2022 15:22:49

%S 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,

%T 42,42,42,49,56,56,56,56,56,56,56,64,72,72,72,72,72,72,72,72,81,90,90,

%U 90,90,90,90,90,90,90,100,110,110,110,110,110,110,110,110,110,110,121

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

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

%H Reinhard Zumkeller, <a href="/A056942/b056942.txt">Table of n, a(n) for n = 0..10000</a>

%F 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).

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

%e X.XX

%e X..X

%e XXXX

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

%o (Haskell)

%o a056942 n = a056942_list !! n

%o a056942_list = concatMap

%o (\x -> (x ^ 2) : (take x $ repeat (x * (x + 1)))) [0..]

%o -- _Reinhard Zumkeller_, Feb 11 2014

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

%K easy,nice,nonn

%O 0,3

%A _Henry Bottomley_, Jul 13 2000