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A309764
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a(0)=0; for n > 0, a(n) = a(n-1) + d * n, where initially d = 1 but changes sign whenever n = |a(k)|, 0 < k < n.
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1
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0, 1, 3, 0, -4, -9, -15, -22, -30, -21, -11, -22, -34, -47, -61, -46, -30, -13, 5, 24, 44, 23, 45, 22, 46, 71, 97, 124, 152, 181, 151, 120, 88, 55, 89, 124, 160, 197, 235, 274, 314, 355, 397, 440, 396, 441, 395, 442, 490, 539, 589, 640, 692, 745, 799, 744, 688, 631, 573, 514
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OFFSET
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0,3
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COMMENTS
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The sequence oscillates erratically from above to below zero. For n <= 10^9 it changes sign 1823 times, with the longest gap between changes occurring from n = 312214179 to n = 703640490, a run of 391426311 negative values. The maximum run without d changing sign starts at n = 726820728 and goes for 43207 values. The longest run of consecutive sign changes in d is four, which occurs four times, at n = 21, 44, 395, and 439. In total d changes sign 675277 times. The largest positive a(n) is for n = 765504445 with a(n) = 496154732190565. The largest negative a(n) is for n = 951481398 with a(n) = -618690200847941.
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LINKS
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EXAMPLE
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a(2) = 3, d = 1. For n = 3 as n = |a(2)|, d changes to d = -1. Therefore a(3) = a(2) + d * 3 = 0.
a(8) = -30, d = -1, and a(5) = -9. For n = 9 as n = |a(5)|, d changes to d = 1. Therefore a(9) = a(8) + d * 9 = -21.
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MATHEMATICA
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s = {0}; d = 1; Do[d *= If[MemberQ[s, n] || MemberQ[s, -n], -1, 1]; AppendTo[s, s[[-1]] + d*n], {n, 1, 60}]; s (* Amiram Eldar, Aug 16 2019 *)
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PROG
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(PARI) s=0; v=0; d=1; for (n=1, mx=60, print1 (v, ", "); if (bittest(s, n), d=-d); v+=d*n; w=abs(v); if (w<=mx, s=bitor(s, 2^w))) \\ Rémy Sigrist, Aug 16 2019
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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