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A064784
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Difference between n-th triangular number t(n) and the largest square <= t(n).
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6
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0, 2, 2, 1, 6, 5, 3, 0, 9, 6, 2, 14, 10, 5, 20, 15, 9, 2, 21, 14, 6, 28, 20, 11, 1, 27, 17, 6, 35, 24, 12, 44, 32, 19, 5, 41, 27, 12, 51, 36, 20, 3, 46, 29, 11, 57, 39, 20, 0, 50, 30, 9, 62, 41, 19, 75, 53, 30, 6, 66, 42, 17, 80, 55, 29, 2, 69, 42, 14, 84, 56, 27, 100, 71, 41, 10, 87, 56
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OFFSET
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1,2
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COMMENTS
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The second differences of a(n) - (a(n)-a(n-1))-(a(n-1)-a(n-2)) - give 2, -2, -1, 6, -6, -1, -1, 12, -12, -1, 16, -16, -1 ... 82k+2, 82k-2, -1, 82k+6, 82k-6, -1, -1, 82k+12, 82k-12, -1, 82k+16, -82k-16, -1, 82k+20, -82k-20, -1, -1, 82k+26, -82k-26, -1, 82k+30, -82k-30, -1, -1, 82k+36, -82k-36, -1, 82k+40, -82k-40, -1, 82k+44, -82k-44, -1, -1, 82k+50, -82k-50, -1, 82k+54, -82k-54, -1, -1, 82k+60, -82k-60, -1, 82k+64, -82k-64, -1, -1, 82k+70, -82k-70, -1, 82k+74, -82k-74, -1, 82k+78, -82k-78, -1, -1, ...
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LINKS
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J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, Exper. Math. 11 (2002), 437-446.
J. C. Lagarias, E. M. Rains and N. J. A. Sloane, The EKG sequence, arXiv:math/0204011 [math.NT], 2002.
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FORMULA
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a(n) = n*(n+1)/2 - floor(sqrt(n*(n+1)/2))^2.
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EXAMPLE
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n = 5: A000217(5) = 28, largest square below that is 25, so a(5) = 28 - 25 = 3.
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MAPLE
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seq(n*(n+1)/2-floor(sqrt(n*(n+1)/2))^2, n=0..100);
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MATHEMATICA
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#-Floor[Sqrt[#]]^2&/@Accumulate[Range[100]] (* Harvey P. Dale, Oct 15 2014 *)
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PROG
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(PARI) { default(realprecision, 100); for (n=1, 1000, t=n*(n + 1)/2; a=t - floor(sqrt(t))^2; write("b064784.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 25 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 20 2001
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EXTENSIONS
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STATUS
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approved
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