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A056944
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Amount by which used area of rectangle needed to enclose a non-touching spiral of length n on a square lattice exceeds unused area.
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6
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0, 1, 2, 2, 2, 4, 3, 2, 4, 6, 4, 2, 4, 6, 8, 5, 2, 4, 6, 8, 10, 6, 2, 4, 6, 8, 10, 12, 7, 2, 4, 6, 8, 10, 12, 14, 8, 2, 4, 6, 8, 10, 12, 14, 16, 9, 2, 4, 6, 8, 10, 12, 14, 16, 18, 10, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 11, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 12, 2, 4, 6, 8, 10, 12, 14, 16
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| m (when n is m-th triangular number) followed by m even numbers from 2 through 2m.
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FORMULA
| a(n) =2n-floor[(sqrt(8n+1)-1)/2]*ceiling[(sqrt(8n+1)-1)/2] =2n-A002024(n)*A003056(n) =2n-A056942(n) =n-A056943(n). If n=t(t+1)/2 then a(n)=t; if n=t(t+1)/2+k with 0<k <= t then a(n)=2k.
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EXAMPLE
| a(9)=6 since spiral is as marked by 9 X's in 4*3=12 rectangle, with 12-9=3 spaces unused and a used-unused difference of 9-3=6:
X.XX
X..X
XXXX
As a triangle, the first few rows are: 1; 2, 2; 2, 4, 3; 2, 4, 6, 4; 2, 4, 6, 8, 5; 2, 4, 6, 8, 10, 6; 2, 4, 6, 8, 10, 12, 7; ... (= reversal of triangle A143595). Row sums = n^2 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008]
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CROSSREFS
| Cf. A002024, A003056, A056942, A056943.
A143595 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 26 2008]
Sequence in context: A064025 A054709 A121806 * A194319 A050493 A085454
Adjacent sequences: A056941 A056942 A056943 * A056945 A056946 A056947
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KEYWORD
| easy,nonn,nice
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jul 13 2000
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