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A010751
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Up once, down twice, up three times, down four times, ...
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9
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0, 1, 0, -1, 0, 1, 2, 1, 0, -1, -2, -1, 0, 1, 2, 3, 2, 1, 0, -1, -2, -3, -2, -1, 0, 1, 2, 3, 4, 3, 2, 1, 0, -1, -2, -3, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -5, -4
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refs;
listen;
history;
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internal format)
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OFFSET
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0,7
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COMMENTS
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Also x-coordinates of a point moving in a spiral rotated by Pi/4, with y-coordinates given by A305258. - Hugo Pfoertner, May 29 2018
This sequence is also obtained by reading alternately in ascending or descending way the antidiagonals of the array defined as A(i, j) = floor((j - i + 1)/2) (see Example). - Stefano Spezia, Jan 02 2022
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LINKS
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FORMULA
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a(n) = x + 1 - (sign(x(2x+1) - y(2y+1)))*(n-2x^2-3x-1) where x = floor((-1-sqrt(1+8n))/4), y = -floor((1-sqrt(1+8n))/4), sign(x) = abs(x)/x when x is not 0 and sign(0) = 0, floor(x) is the greatest integer less than or equal to x, sqrt(x) is the principal square root of x and abs(x) is the absolute value (or magnitude) of x. - Mark Spindler, Mar 25 2004
a(A007590(k)) = a(floor(k^2 / 2)) = 0.
a(A000384(k)) = a(binomial(2 * k, 2)) = k, a new maximum so far.
a(A014105(k)) = a(binomial(2 * k + 1, 2)) = -k, a new minimum so far.
(End)
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EXAMPLE
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The array A begins with:
0 1 1 2 2 3 3 ...
0 0 1 1 2 2 3 ...
-1 0 0 1 1 2 2 ...
-1 -1 0 0 1 1 2 ...
-2 -1 -1 0 0 1 1 ...
-2 -2 -1 -1 0 0 1 ...
...
(End)
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MATHEMATICA
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n=(the index); x = -1; y = 0; While[n != 0, While[y != x && n != 0, y--; n-- ]; While[y != -x && n != 0, n--; y++ ]; x-- ]; Print[ -y] (* provided by Gregory Puleo *)
n = (the index); a = Floor[(-1 - Sqrt[1 + 8* n])/4]; b = -Floor[(1 - Sqrt[1 + 8*n])/4]; a + 1 - Sign[a*(2*a + 1) - b*(2*b + 1)]*(n - 2*a^2 - 3*a - 1) (* Mark Spindler, Mar 25 2004 *)
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PROG
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(PARI) step=-1; print1(x=0, ", "); for(stride=1, 12, step=-step; for(k=1, stride, print1(x+=step, ", "))) \\ Hugo Pfoertner, Jun 02 2018
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CROSSREFS
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KEYWORD
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sign,easy,changed
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AUTHOR
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David Berends (dave(AT)pgt.com)
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STATUS
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approved
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