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A056942
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Area of rectangle needed to enclose a non-touching spiral of length n on a square lattice.
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2
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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
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OFFSET
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0,3
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COMMENTS
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m^2 (when n is m-th triangular number) followed by m copies of m-th pronic [m(m+1)].
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LINKS
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Table of n, a(n) for n=0..66.
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FORMULA
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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).
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EXAMPLE
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a(9)=12 since spiral is as marked by 9 X's in 4*3=12 rectangle:
X.XX
X..X
XXXX
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MATHEMATICA
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ar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2}, Floor[c]Ceiling[c]]; Array[ar, 70, 0] (* From Harvey P. Dale, Feb 27 2012 *)
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CROSSREFS
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Cf. A000217, A000290, A002024, A002378, A002620, A003056, A038759, A056943, A056944.
Sequence in context: A049066 A151688 A159276 * A115947 A061228 A070229
Adjacent sequences: A056939 A056940 A056941 * A056943 A056944 A056945
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Henry Bottomley, Jul 13 2000
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STATUS
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approved
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