This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A010751 #62 Aug 05 2025 15:26:21 %S A010751 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, %T A010751 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, %U A010751 -3,-2,-1,0,1,2,3,4,5,6,5,4,3,2,1,0,-1,-2,-3,-4,-5,-6,-5,-4 %N A010751 Up once, down twice, up three times, down four times, ... %C A010751 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 %C A010751 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 %H A010751 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a> %F A010751 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 %F A010751 From _David A. Corneth_, Jun 02 2018: (Start) %F A010751 a(A007590(k)) = a(floor(k^2 / 2)) = 0. %F A010751 a(A000384(k)) = a(binomial(2 * k, 2)) = k, a new maximum so far. %F A010751 a(A014105(k)) = a(binomial(2 * k + 1, 2)) = -k, a new minimum so far. %F A010751 (End) %F A010751 a(n) = (-1)^A002024(n+1)*(A007590(A002024(n+1))-n). - _William McCarty_, Jul 30 2021 %e A010751 From _Stefano Spezia_, Jan 02 2022: (Start) %e A010751 The array A begins with: %e A010751 0 1 1 2 2 3 3 ... %e A010751 0 0 1 1 2 2 3 ... %e A010751 -1 0 0 1 1 2 2 ... %e A010751 -1 -1 0 0 1 1 2 ... %e A010751 -2 -1 -1 0 0 1 1 ... %e A010751 -2 -2 -1 -1 0 0 1 ... %e A010751 ... %e A010751 (End) %t A010751 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 *) %t A010751 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 *) %o A010751 (PARI) step=-1;print1(x=0,", ");for(stride=1,12,step=-step;for(k=1,stride,print1(x+=step,", "))) \\ _Hugo Pfoertner_, Jun 02 2018 %o A010751 (Python) %o A010751 from math import isqrt %o A010751 def A010751(n): return n-(m**2>>1) if (m:=isqrt(n+1<<3)+1>>1)&1 else (m**2>>1)-n # _Chai Wah Wu_, Jun 08 2025 %Y A010751 Cf. A000384, A002024, A007590, A014105, A305258. %K A010751 sign,easy,tabl %O A010751 0,7 %A A010751 David Berends (dave(AT)pgt.com)