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A056943
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Unused 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, 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
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OFFSET
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0,5
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LINKS
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FORMULA
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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.
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EXAMPLE
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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
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MATHEMATICA
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uar[n_]:=Module[{c=(Sqrt[8n+1]-1)/2}, Floor[c]Ceiling[c]-n]; Array[ uar, 80, 0] (* Harvey P. Dale, Oct 29 2015 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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