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 A220098 #28 Jan 11 2025 18:52:26
%S A220098 1,2,1,2,3,2,3,4,3,2,3,4,5,4,3,4,5,6,5,4,3,4,5,6,7,6,5,4,5,6,7,8,7,6,
%T A220098 5,4,5,6,7,8,9,8,7,6,5,6,7,8,9,10,9,8,7,6,5,6,7,8,9,10,11,10,9,8,7,6,
%U A220098 7,8,9,10,11,12,11,10,9,8,7,6,7,8,9,10,11,12,13
%N A220098 Manhattan distances between 2n and 1 in the double spiral with positive integers and 1 at the center.
%C A220098 Double spiral begins:
%C A220098 .
%C A220098 82---84---86---88---90---92---94---96---98
%C A220098 |
%C A220098 80 51---53---55---57---59---61---63---65
%C A220098 | | |
%C A220098 78 49 26---28---30---32---34---36 67
%C A220098 | | | | |
%C A220098 76 47 24 11---13---15---17 38 69
%C A220098 | | | | | | |
%C A220098 74 45 22 9 2----4 19 40 71
%C A220098 | | | | | | | | |
%C A220098 72 43 20 7 1 6 21 42 73
%C A220098 | | | | | | | | |
%C A220098 70 41 18 5----3 8 23 44 75
%C A220098 | | | | | | |
%C A220098 68 39 16---14---12---10 25 46 77
%C A220098 | | | | |
%C A220098 66 37---35---33---31---29---27 48 79
%C A220098 | | |
%C A220098 64---62---60---58---56---54---52---50 81
%C A220098 |
%C A220098 99---97---95---93---91---89---87---85---83
%F A220098 abs( a(n) - a(n-1) ) = 1.
%F A220098 From _Thomas Scheuerle_, Jan 07 2025: (Start)
%F A220098 a(n*(n+1)/2 - k) = 1 + a(n*(n-1)/2 + k) with a(0) = 0 and for 0 <= k < n.
%F A220098 a(n) = A053615(A128217(n+1)). (End)
%e A220098 From _Philippe Deléham_, Mar 08 2013: (Start)
%e A220098 As a square array, this begins:
%e A220098 1, 1, 2, 2, 3, 3, 4, 4, 5, ...
%e A220098 2, 3, 3, 4, 4, 5, 5, 6, 6, ...
%e A220098 2, 4, 5, 5, 6, 6, 7, 7, 8, ...
%e A220098 3, 4, 6, 7, 7, 8, 8, 9, 9, ...
%e A220098 3, 5, 6, 8, 9, 9, 10, 10, 11, ...
%e A220098 4, 5, 7, 8, 10, 11, 11, 12, 12, ...
%e A220098 4, 6, 7, 9, 10, 12, 13, 13, 14, ...
%e A220098 5, 6, 8, 9, 11, 12, 14, 15, 15, ..., etc.
%e A220098 As a triangle, this begins:
%e A220098 1
%e A220098 2, 1
%e A220098 2, 3, 2
%e A220098 3, 4, 3, 2
%e A220098 3, 4, 5, 4, 3
%e A220098 4, 5, 6, 5, 4, 3, etc. (End)
%o A220098 (C)
%o A220098 #include <stdio.h>
%o A220098 #define SIZE 20
%o A220098 int grid[SIZE][SIZE];
%o A220098 int direction[] = {0, -1, 1, 0, 0, 1, -1, 0};
%o A220098 main() {
%o A220098 int i, j, x1, y1, x2, y2, stepSize;
%o A220098 int direction1pos=0, direction2pos=4, val;
%o A220098 x1 = y1 = x2 = y2 = SIZE/2;
%o A220098 for (val=grid[y1][x1]=1, stepSize=0; ; ++stepSize) {
%o A220098 if (x1<1 || x1>=SIZE-1 || x2<1 || x2>=SIZE-1) break;
%o A220098 if (y1<1 || y1>=SIZE-1 || y2<1 || y2>=SIZE-1) break;
%o A220098 for (i=stepSize|1; i; ++val,--i) {
%o A220098 x1 += direction[direction1pos ];
%o A220098 y1 += direction[direction1pos+1];
%o A220098 x2 += direction[direction2pos ];
%o A220098 y2 += direction[direction2pos+1];
%o A220098 grid[y1][x1] = val*2;
%o A220098 grid[y2][x2] = val*2+1;
%o A220098 printf("%d, ",abs(x1-SIZE/2)+abs(y1-SIZE/2));
%o A220098 }
%o A220098 direction1pos = (direction1pos+2) & 7;
%o A220098 direction2pos = (direction2pos+2) & 7;
%o A220098 }
%o A220098 for (i=0; i<SIZE; ++i) {
%o A220098 printf("\n");
%o A220098 for (j=0; j<SIZE; ++j)
%o A220098 printf("%4d ",grid[i][j]);
%o A220098 }
%o A220098 }
%o A220098 (PARI)
%o A220098 step(v, m) = concat(v, vector(m, k, 1+v[#v-k+1]))
%o A220098 a(max_n) = {my(v=[0], k=1); while(#v < max_n+1, v=step(v,k); k++); v[2..max_n+1]} \\ _Thomas Scheuerle_, Jan 07 2025
%o A220098 (PARI)
%o A220098 A053615(n) = if(n<1, 0, sqrtint(n) - A053615(n - sqrtint(n)))
%o A220098 a(n) = A053615(floor( floor( (sqrtint(n*8) + 1)/2 )^2/2 ) + n) \\ _Thomas Scheuerle_, Jan 07 2025
%Y A220098 Cf. A053615, A077043, A128217, A214526.
%K A220098 nonn,easy
%O A220098 1,2
%A A220098 _Alex Ratushnyak_, Dec 04 2012