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A140756
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Count up to k sequence with alternating signs (k always positive).
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3
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1, -1, 2, 1, -2, 3, -1, 2, -3, 4, 1, -2, 3, -4, 5, -1, 2, -3, 4, -5, 6, 1, -2, 3, -4, 5, -6, 7, -1, 2, -3, 4, -5, 6, -7, 8, 1, -2, 3, -4, 5, -6, 7, -8, 9, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13
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history;
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internal format)
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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Regarded as a triangle, T(n,k) = (-1)^{n-k} * k.
a(n) = (-1)^(j+1) * i, where i = n - t*(t+1)/2, j = (t^2 + 3*t + 4)/2 -n, and t = floor((-1 + sqrt(8*n-7))/2). (End)
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EXAMPLE
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Triangle begins:
1;
-1, 2;
1, -2, 3;
-1, 2, -3, 4;
1, -2, 3, -4, 5;
-1, 2, -3, 4, -5, 6;
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MATHEMATICA
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a[n_]:= With[{t=Floor[(-1+Sqrt[8*n-7])/2]}, (-1)^(Binomial[t+2, 2] -n)*(n-Binomial[t+1, 2])];
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PROG
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(Magma) [(-1)^(n+k)*k: k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 21 2023
(SageMath) flatten([[(-1)^(n+k)*k for k in range(1, n+1)] for n in range(1, 13)]) # G. C. Greubel, Oct 21 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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