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A137931
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Sum of the principal diagonals of a 2n X 2n square spiral.
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5
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0, 10, 56, 170, 384, 730, 1240, 1946, 2880, 4074, 5560, 7370, 9536, 12090, 15064, 18490, 22400, 26826, 31800, 37354, 43520, 50330, 57816, 66010, 74944, 84650, 95160, 106506, 118720, 131834, 145880, 160890, 176896, 193930, 212024, 231210, 251520, 272986, 295640
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OFFSET
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0,2
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COMMENTS
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This is concerned with 2n X 2n square spirals of the form illustrated in the Example section.
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LINKS
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Table of n, a(n) for n=0..38.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
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FORMULA
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a(n) = -1 + n + Sum_{k=0..2n} (2k^2 - k + 1) = n -1 +(2*n+1)*(8*n^2-n+3)/3.
a(n) = 2*n^2 + 2*n + (16*n^3 + 2*n)/3 = 2*n*(8*n^2+3*n+4)/3.
G.f.: 2*x*(3*x+5)*(x+1)/(x-1)^4. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
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EXAMPLE
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Example with n = 2:
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7---8---9--10
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6 1---2 11
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5---4---3 12
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16--15--14--13
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a(0) = 2(0)^2 + 2(0) + (16(0)^3 + 2(0))/3 = 0;
a(2) = 2(2)^2 + 2(2) + (16(2)^3 + 2(2))/3 = 56.
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PROG
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(Python) f = lambda n: -1 + n + sum(2*k**2 - k + 1 for k in range(0, 2*n+1))
(Python) a = lambda n: 2*n**2 + 2*n + (16*n**3 + 2*n)/3
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CROSSREFS
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Cf. A137928, A002061. A bisection of A137930.
Sequence in context: A202071 A281207 A228888 * A053493 A198833 A268462
Adjacent sequences: A137928 A137929 A137930 * A137932 A137933 A137934
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KEYWORD
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nonn,easy
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AUTHOR
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William A. Tedeschi, Feb 29 2008
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STATUS
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approved
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