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A137932
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Terms in an n X n spiral that do not lie on its principal diagonals.
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1
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0, 0, 0, 4, 8, 16, 24, 36, 48, 64, 80, 100, 120, 144, 168, 196, 224, 256, 288, 324, 360, 400, 440, 484, 528, 576, 624, 676, 728, 784, 840, 900, 960, 1024, 1088, 1156, 1224, 1296, 1368, 1444, 1520, 1600, 1680, 1764, 1848, 1936, 2024, 2116, 2208, 2304, 2400, 2500, 2600, 2704, 2808
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| The count of terms not on the priciple diagonals is always even.
The last digit is the repeating pattern 0,0,0,4,8,6,4,6,8,4, which is palindromic if the leading 0s are removed, 4864684.
The sum of the last digits is 40, which is the count of the pattern times 4.
A 4 X 4 spiral is the only spiral, aside from a 0x0, whose count of terms that do not lay on its pricipal diagonals equal the count of terms that do [A137932(4) = A042948(4)] making the 4 X 4 the "perfect spiral."
Yet another property is mod(A137932(n),A042948(n)) = 0 iff n is even. This is a large family that includes the 4 X 4 spiral.
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FORMULA
| a(n) = n^2 - (2n - mod(n,2)) = n^2 - A042948(n)
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EXAMPLE
| a(0) = 0^2 - (2(0) - mod(0,2)) = 0
a(3) = 3^2 - (2(3) - mod(3,2)) = 4
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PROG
| (Python) a = lambda n: n**2 - (2*n - (n%2))
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CROSSREFS
| Cf. A042948.
Sequence in context: A160740 A181823 A046059 * A140466 A161226 A022560
Adjacent sequences: A137929 A137930 A137931 * A137933 A137934 A137935
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KEYWORD
| nonn,easy
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AUTHOR
| William A. Tedeschi (fynmun(AT)hotmail.com), Feb 29 2008
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