A009000 -id:A009000 - cp's OEIS frontend

A309500 The minimal number of steps to return to the origin for a self-avoiding walk with step-length n on a 2D plane where at each step the walk must go to an unvisited point with integer coordinates as close as possible to the origin.

4, 4, 4, 4, 36, 4, 4, 4, 4, 36, 4, 4, 122, 4, 36, 4, 198, 4, 4, 36, 4, 4, 4, 4, 436, 122, 4, 4, 632, 36, 4, 4, 4, 198, 36, 4, 694, 4, 122, 36

Offset: 1

Scott R. Shannon, Aug 05 2019

Consider a walk on a 2D plane starting at the (0,0) origin. We introduce the following rules for the walk: 1. Each step is of length n. 2. The walk can only step to points with integer x and y coordinates. 3. The walk cannot go to a point already visited, nor can it backtrack on its very first step from the origin. 4. At each step the walk must always go to a point which is as close as possible to the origin. Given these rules, what is the minimum number of steps required for the walk to return to the origin? For step length n the sequence a(n) is the minimum number of steps.
For step lengths n which are not hypotenuses of any Pythagorean triple the solution is 4 as the walk will simply trace out a square e.g. step up, then right (or left), then down, then left (or right) back to the origin.  This is true for n=1,2,3,4. But for step length 5 if the initially step is to (0,5) then by the rules the closest point to the origin for the next step now becomes (3,1) - the step must take the hypotenuse of a (3,4,5) triangle as clearly (3,1) is closer to the origin than (5,5). From here the walk must continue to pick the next point as close as possible to the origin which is 5 units away from its current coordinate, either directly left-right-up-down or by moving along the hypotenuse of a (3,4,5) triangle. Also note for n=5, and any step length which is the hypotenuse of a Pythagorean triple, the walk can also take a first step to (4,3) - this can result in a totally different path that may, or may not, lead back to the origin in fewer steps.
If the walk visits a point either on the x=0 or y=0 axis, or along the line y=x or y=-x, then it is likely that for the next step two points will be available which are equidistant from the origin. As we do not know which will lead to the minimum length walk we are searching for, the walker must explore both paths. Although the values of a(n) for the primitive Pythagorean triples are not particularly large, the total number of paths that must be recursively searched to find these minimum values increases extremely rapidly - to find a(37) required searching at least 400 million different paths, these being cut off by the walk either returning to the origin in fewer steps than the current minimum path, or by their step count surpassing the current minimum path. Due to the explosive increase in the total number of paths the exact value has only been determined for n=5. Assuming a first step to (0,5) or (4,3) the total number of paths is 26421. A n=5 random walk therefore has about a 1 in 2400 chance of taking one of the minimum 36 step walks.
For a(n) up to 40 it is found that for all n corresponding to Pythagorean triple hypotenuses that there are multiple paths corresponding to the minimum path value e.g. for n=5 there are 11 different paths all of which return to the origin after 36 steps. For n=37 there are 2048 different paths, once again showing the rapid increase in the total paths that must be explored to find the minimum.
Integer multiples of the hypotenuses of the primitive Pythagorean triples result in the same values for a(n) - simply scaling up the step length does not change the minimum path. However this is not necessarily true if the resulting hypotenuse has more than one triple e.g. n=25. The addition choice of a second Pythagorean triple greatly increases the number of paths to explore: finding a(26), which is the same as a(13), required searching about 550 paths, but a(25) requires searching at least 37 million paths.
It seems plausible that the path could be trapped by surrounding visited points during a very long walk and thus never be able to return to the origin. It is unknown if this can occur.
The author thanks Zachary Shannon and David Gundersen for discussions and efficiency improvements to the search program.
From Bert Dobbelaere, Aug 18 2019: (Start)
All terms are even. For odd n, consider the grid colored like a chessboard. It follows from the parity relation of Pythagorean triples that the destination of each step is of different color than the source. Hence each closed walk takes an even number of steps. For even n, no odd coordinates can be encountered, so we can divide the problem size by 2 until we are back in the odd case.
a(41) <= 1244 ; a(53) = 1528. (End)
From Scott R. Shannon, Oct 23 2019: (Start)
a(61) <= 3698 ; a(73) = 3080 ; a(89) = 1034 ; a(97) = 4734. These results indicate that for hypotenuse lengths n which have legs of approximately the same length the number of search paths to find the minimum is relatively small. For lengths which have legs of uneven length, e.g. 41, the number of search paths becomes extremely large; n = 41 requires searching at least 10^12 paths. (End)

			a(1) = 4. Path: (0,0)->(0,1)->(1,1)->(1,0)->(0,0).
a(5) = 36. Path: (0,0)->(4,3)->(1,-1)->(-3,2)->(0,-2)->(0,3)->(-3,-1)->(2,-1)->(-2,2)->(1,-2)->(1,3)->(-2,-1)->(2,2)->(-1,-2)->(-1,3)->(3,0)->(-2,0)->(2,3)->(-1,-1)->(3,2)->(3,-3)->(0,1)->(0,-4)->(-3,0)->(2,0)->(-2,3)->(-2,-2)->(2,1)->(-1,-3)->(-1,2)->(2,-2)->(-2,1)->(1,-3)->(1,2)->(4,-2)->(0,-5)->(0,0). Note this path is one of eleven with a total path length of 36. Five of those eleven paths step to (-3,2) on the third step while the other six step to (-2,3). The first step to the point (4,3) is required - a first step to (0,5) results in a minimum path length of 40.

Scott R. Shannon, Walk path for n=5. For this, and other images, the line colors are scaled from red to violet to show the relative step at which the point was visited. A white square marks the origin, a red square the first step point, and a violet square the last point before return to the origin, the path of which is marked with a white line. In this image only the smaller yellow squares indicate the points where the path had a choice of 2 points to go to which were equidistant from the origin. This is one of eleven different paths which return to the origin in 36 steps.
Scott R. Shannon, Visited points for n=5. These points show a pattern which is repeated in the other minimum step walks - the majority of points about half a step length away from the origin are visited. This eventually forces the walk to move away from the origin at which point a coordinate one step length away from the origin is finally visited.
Scott R. Shannon, Walk path for n=13.
Scott R. Shannon, Walk path for n=17.
Scott R. Shannon, Walk path for n=25.
Scott R. Shannon, Visited points for n=25. This clearly shows the tendency for the minimal path to cover the points half a step length away from the origin before visiting the last point which is one step length away from the origin.
Scott R. Shannon, Walk path for n=29. As the Pythagorean triple for 29 is (20,21,29) the diagonal steps are almost at 45 degrees to the axis which causes this path to form, in the authors opinion, a very aesthetically pleasing pattern. The next step length with legs one unit apart which has no other Pythagorean triples, and is thus likely to form a similar pattern, is 5741. Unfortunately the number of search paths to find the minimum in this instance would be astronomically large.
Scott R. Shannon, Visited points for n=29.
Scott R. Shannon, Walk path for n=37.
Scott R. Shannon, Visited points for n=37.
Scott R. Shannon, Walk path for n=53.
Scott R. Shannon, Walk path for n=73.
Scott R. Shannon, Walk path for n=89.
Scott R. Shannon, Walk path for n=97.
Wikipedia, Pythagorean Triples.

Cf. A020882, A009000.

For step length n, also equal to the hypotenuse of one Pythagorean triple (a,b,n) where a>b, the first step of the walk can be either along the y axis or along the hypotenuse diagonal in the positive x-y direction (all other direction choices are equivalent by rotation/reflection). In the former case the first point will be (0,n), and thus the second point will be (b,n-a). From here a step directly left to (-(n-b),n-a) or diagonally down and left to (-(a-b),-(b-(n-a)) result in points equidistant from the origin - both are sqrt(3n^2-2bn-2an) from (0,0). So after only two steps two valid branches arise both of which must be explored. If instead the first step is along the hypotenuse to (a,b) then using simple algebra one can show that the next step will be to (-(n-a),b) if n>2b, else it will be to (a-b,-(a-b)) for n<2b. Note that in the former case the path (0,0)->(a,b)->(-(n-a),b) is rotationally symmetric to the path (0,0)->(0,n)->(b,n-a), so for step lengths n with n>2b only one of these options needs to be explored.

A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.

Original entry on oeis.org

15, 25, 30, 35, 39, 50, 51, 55, 60, 65, 65, 70, 75, 75, 78, 85, 85, 87, 91, 95, 100, 102, 105, 110, 111, 115, 119, 120, 123, 125, 130, 130, 135, 140, 143, 145, 145, 150, 150, 155, 156, 159, 165, 169, 170, 170, 174, 175, 175, 182, 183, 185, 185, 187, 190, 195, 195

Offset: 1

Felix Huber, Aug 13 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2) is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
A101930(n) gives the total number of Pythagorean triples <= 10^n. The percentage of triangles in this sequence increases continuously:
               number of terms <= h     total number of
       h       in this sequence         hypotenuses <= h      percentage
      10                 0                    2                  0.0 %
     100                21                   52                 40.4 %
    1000               514                  881                 58.3 %
   10000              8629                12471                 69.2 %
  100000            122431               161436                 75.8 %

			The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 15 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A009000.

Cf. A101930, A366428, A380072, A386308, A386309, A386943.

A386307:=proc(N) # To get all hypotenuses <= N
    local i,l,m,u,v,r,x,y,z;
    l:={};
    m:={};
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            x:=min(2*u*v,u^2-v^2);
            y:=max(2*u*v,u^2-v^2);
            z:=u^2+v^2;
            m:=m union {[z,y,x]};
            if gcd(u,v)=1 and is(u-v,odd) then
                l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
            fi
        od
    od;
    r:=l minus m;
    return seq(r[i,1],i=1..nops(r));
end proc;
A386307(1000);

a(n) = sqrt(A386308(n)^2 + A386309(n)^2).

{A009000(n)} = {a(n)} union {A020882(n)} union {A386943(n)}.

A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.

Original entry on oeis.org

10, 20, 26, 34, 40, 45, 50, 52, 58, 68, 74, 80, 82, 90, 100, 104, 106, 116, 117, 122, 125, 130, 130, 136, 146, 148, 153, 160, 164, 170, 170, 178, 180, 194, 200, 202, 208, 212, 218, 225, 226, 232, 234, 244, 245, 250, 250, 260, 260, 261, 272, 274, 290, 290, 292, 296

Offset: 1

Felix Huber, Aug 24 2025

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
A101930(n) gives the total number of Pythagorean triples <= 10^n.
               number of terms <= h     total number of
       h       in this sequence         hypotenuses <= h      percentage
      10                 1                    2                 50.0 %
     100                15                   52                 28.8 %
    1000               209                  881                 23.7 %
   10000              2249                12471                 18.0 %
  100000             23086               161436                 14.3 %

			The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 10 is a term.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Subsequence of A009000.

Cf. A020882, A101930, A366428, A380072, A386307, A386944, A386945.

A386943:=proc(N) # To get all hypotenuses <= N
    local i,l,u,v;
    l:=[];
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            if gcd(u,v)>1 or is(u-v,even) then
                l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
            fi
        od
    od;
    l:=sort(l);
    return seq(l[i,1],i=1..nops(l));
end proc;
A386943(296);

a(n) = sqrt(A386944(n)^2 + A386945(n)^2).

{A009000(n)} = {a(n)} union {A020882(n)} union {A386307(n)}.

A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 0, 0, 0, 6, 0, 0, 5, 0, 9, 0, 8, 0, 0, 12, 0, 0, 0, 0, 7, 10, 0, 0, 20, 18, 0, 0, 0, 16, 21, 0, 12, 0, 15, 24, 9, 0, 0, 0, 27, 0, 0, 0, 0, 14, 24, 20, 28, 0, 33, 0, 0, 40, 0, 36, 11, 0, 0, 0, 16, 0, 0, 32, 0, 42, 0, 0, 48, 24, 21, 0, 0, 30, 0, 48, 0, 18, 0, 0, 13, 0, 60, 0, 39, 54

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) = 3 as the right triangle with sides (3, 4, 5) has hypotenuse n = 5 smallest side a(5) = 3. This is the smallest side a right triangle with integer sides and hypotenuse 5 can have. - _David A. Corneth_, Apr 10 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108708.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_]:=Block[{k=n-1,m=Sqrt[n/2],a},While[k>m&&!IntegerQ[(a=Sqrt[n^2-k^2])],k--];If[k<=m,0,a]];Table[f[n],{n,90}]

first(n) = {my(lh = List(), res = vector(n, i, oo)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = vecmin([res[i*(u^2+v^2)], i*(u^2 - v^2), i*2*u*v]))))); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

Extended by Ray Chandler, Dec 20 2011

A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 12, 0, 12, 0, 15, 0, 0, 16, 0, 0, 0, 0, 24, 24, 0, 0, 21, 24, 0, 0, 0, 30, 28, 0, 35, 0, 36, 32, 40, 0, 0, 0, 36, 0, 0, 0, 0, 48, 45, 48, 45, 0, 44, 0, 0, 42, 0, 48, 60, 0, 0, 0, 63, 0, 0, 60, 0, 56, 0, 0, 55, 70, 72, 0, 0, 72, 0, 64, 0, 80, 0, 0, 84, 0, 63, 0

Offset: 1

Sébastien Dumortier, Jun 20 2005

nonn

			a(5) is 4 as the maximum side (other than the hypotenuse) a right triangle with integer sides and hypotenuse 5 can have.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A083025, A046083, A046084, A009000, A009003, A108707.

A046080 gives the number of Pythagorean triangles with hypotenuse n.

f[n_] := Block[{k = n - 1, m = Sqrt[n/2]}, While[k > m && !IntegerQ[Sqrt[n^2 - k^2]], k-- ]; If[k <= m, 0, k]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Jun 21 2005 *)

first(n) = {my(lh = List(), res = vector(n)); for(u = 2, sqrtint(n), for(v = 1, u, if (u^2+v^2 > n, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, n, if (i*(u^2+v^2) > n, break); listput(lh, i*(u^2+v^2)); res[i*(u^2+v^2)] = max(res[i*(u^2+v^2)], max(i*(u^2 - v^2), i*2*u*v)); ); ); ); ); for(i = 1, n, if(res[i] == oo, res[i] = 0)); res } \\ David A. Corneth, Apr 10 2021, adapted from A009000

More terms from Robert G. Wilson v, Jun 21 2005

A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.

Original entry on oeis.org

560, 736, 1541, 3201, 5461, 6816, 7400, 9976, 11041, 11408, 13333, 14981, 15408, 15841, 19521, 21000, 21505, 25761, 28616, 30401, 41536, 45141, 50440, 51221, 52008, 54405, 56856, 61920, 63656, 65416, 69008, 75525, 76480, 81345, 82336, 85345, 87381, 89441

Offset: 1

Jonathan Vos Post, Dec 25 2007

For sums of two positive octagonal numbers, see A136345. This is to octagonal numbers A000567 as A089982 is to triangular numbers A000217, as A009000 is to squares A000290, as A136117 is to pentagonal numbers A000326, as A133215 is to hexagonal numbers A000384, and as A117104 is to heptagonal numbers A000566. If Oc(a) + Oc(b) = Oc(c) then a(3a-2) + b(3b+2) = c(3c+2), so solving the quadratic equations for c we have (when an integer): c = (2 + sqrt(4 + 36a^2 + 36b^2 - 24a - 24b))/6.

			Where Oc(n) = A000567(n) = n-th octagonal number:
a(1) = 560 = Oc(14) = 280 + 280 = Oc(10) + Oc(10).
a(2) = 736 = Oc(16) = 560 + 176 = Oc(14) + Oc(8).
a(3) = 1541 = Oc(23) = 1408 + 133 = Oc(22) + Oc(7).
a(4) = 3201 = Oc(33) = 2465 + 736 = Oc(29) + Oc(16).
a(5) = 5461 = Oc(43) = 2821 + 2640 = Oc(31) + Oc(30).

B. D. Swan, Table of n, a(n) for n = 1..1800
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A009000, A089982, A136117, A136345.

Module[{nn=300,ono},ono=PolygonalNumber[8,Range[nn]];Union[Select[ Total/@ Tuples[ono,2],MemberQ[ono,#]&]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 26 2019 *)

A000567 INTERSECTION {A000567(i) + A000567(j), i, j > 0}. {i*(3*i-2)} INTERSECTION {i*(3*i-2) + j(3*j-2), i > 0}.

Corrected and edited by B. D. Swan (bdswan(AT)gmail.com), Dec 20 2008

A307448 List of pairs of coordinates (x,y) of the visited points for a self-avoiding walk with an incrementing step length confined to one quadrant of a 2D plane where at each step the walk must go to an unvisited point with integer coordinates as close as possible to the origin.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 2, 4, 2, 0, 2, 5, 8, 5, 1, 5, 9, 5, 0, 5, 10, 5, 10, 16, 10, 4, 5, 16, 5, 2, 5, 17, 5, 1, 5, 18, 5, 0, 5, 19, 17, 3, 17, 24, 17, 2, 17, 25, 17, 1, 2, 21, 26, 11, 26, 38, 26, 10, 5, 30, 23, 6, 23, 37, 23, 5, 23, 38, 7, 8, 42, 8, 6, 8, 43, 8, 5, 8, 44, 8, 4, 8, 45, 8, 3, 8, 46, 8, 2, 8, 47

Offset: 1

Scott R. Shannon, Aug 09 2019

Consider a walk on a 2D plane starting at the (0,0) origin which is confined to the positive x and y quadrant, i.e., x >= 0, y >= 0. We introduce the following rules for the walk: 1. The step length, initially 1, increments by 1 after each step. 2. The walk can only step to grid points with integer x and y coordinates. 3. The walk cannot go to a grid point already visited, nor can it backtrack on its very first step from the origin. 4. At each step the walk must always go to a point which is as close as possible to the origin. Given these rules, and a first step of length 1 to (0,1), this sequence gives the (x,y) coordinate points that are visited by the walk.
The sequence has 33305 pairs of coordinates. On the 33304th step the walk visits point (498,498) and now two points, (33803,498) and (498,33803), are both unvisited and available for the next step, thus beyond this point the walk/sequence is not uniquely defined.
For step lengths > 100 the closest approach to the origin is at step 2032 which visits point (6,0). The smallest ratio of distance-to-origin to step length occurs at step 27628 which visits point (7,15). The largest visited x and y coordinates are 42165 and 38631 respectively.
One finds there are long lines of adjacent visited points, differing by two step lengths, which are due to the path moving outward and then back inward along the x and y axial directions. These continue until the innermost point encounters an axis or a Pythagorean diagonal step is found which is closer to the origin. These adjacent points are equivalent to the points forming the 'concentric circles' of the Recamán sequence A005132 when plotted as a spiral. There is also a general clustering of the points towards the axis as well as the y=x diagonal.
This sequence raises the question - is there a walk that ultimately returns to the origin? Currently this is unknown. Beyond the 33304th point a recursive search must be perform on each branching path. Investigating beyond the first branching point show that the next branch occurs at step 200477, at point (121959,121959), followed by a series of branches at step lengths 472952, 730405, 974413.

			The first 12 steps are along the x-y axial directions. But on the 13th step the point (5,16) is available and the closest possible to the origin - this is visited by stepping along the hypotenuse of a (5,12,13) Pythagorean triangle from the point (10,4).

Scott R. Shannon, Table of n, a(n) for n = 1..66610
Scott R. Shannon, Plot of points with x and y values less than 5000. In this and other images the pixel colors are scaled from red to violet to show the relative step at which the point was visited. Each pixel is one point. The final (498,498) point is plotted with a larger white square for clarity.
Scott R. Shannon, Plot of all the sequence points. Due to the maximum x and y visited coordinate being much larger than the image size each pixel covers multiple points. For clarity each visited point is shown with a multi-pixel square.
Scott R. Shannon, Plot of the walk steps. A small red dot near the origin is the final (498,498) point.
Scott R. Shannon, Step visit to points for x<100 and y<100. This shows the step number at which each point was visited for points with x and y < 100. Best viewed using a text editor with word-wrapping turned off. The origin is marked as -1.
Wikipedia, Pythagorean Triples.

Cf. A020882, A009000, A005132.

A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

Original entry on oeis.org

1, 2, 8, 27, 39, 54, 73, 98, 133, 186, 273, 426, 709, 1250, 2305, 4386, 8517, 16746, 33169, 65978, 131557, 262674, 524865, 1049202, 2097829, 4195034, 8389393, 16778058, 33555333, 67109826, 134218753, 268436546, 536872069, 1073743050, 2147484945, 4294968666, 8589936037, 17179870706

Offset: 0

Scott R. Shannon, Sep 08 2021

nonn

This sequence uses the same rules as A347594 except here all numbers must be unique. Up to 10^5 terms all terms are larger than the previous term; it is unknown if this holds for all terms as n->infinity.

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 8 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 8 = 16 = 4^2. Note that although 8 + 1 = 9 = 3^2, 1 cannot be chosen as a(0) = 1.
a(3) = 27 as a(2)^2 + 3^2 = 64 + 9 = 73 and 73 + 27 = 100 = 10^2.  Note that although 73 + 8 = 81 = 9^2, 8 cannot be chosen as a(2) = 8.
a(4) = 39 as a(3)^2 + 4^2 = 729 + 16 = 745, and 745 + 39 = 784 = 28^2 is the next smallest square.

Cf. A347594, A000290, A103605, A009000, A005408, A347754.

A349119 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that |n - a(n-1)| + a(n) is a square.

Original entry on oeis.org

1, 4, 2, 3, 8, 6, 9, 7, 15, 10, 16, 11, 24, 5, 27, 13, 22, 20, 14, 31, 25, 12, 26, 33, 40, 21, 44, 19, 55, 23, 18, 36, 32, 35, 48, 51, 34, 46, 17, 42, 47, 30, 37, 43, 63, 82, 28, 45, 61, 52, 62, 38, 50, 78, 57, 79, 41, 65, 29, 70, 39, 59, 97, 66, 98, 67, 80, 68, 49, 101, 69, 119, 53, 124, 71, 60

Offset: 0

Scott R. Shannon, Nov 08 2021

			a(1) = 4 as |1 - a(0)| = |1 - 1| = 0, and 0 + 4 = 4 = 2^2 is the next smallest square. Note a(1) cannot be 1 as a(0) = 1.
a(4) = 8 as |4 - a(3)| = |4 - 3| = 1, and 1 + 8 = 9 = 3^2 is the next smallest square. Note a(4) cannot be 3 as a(3) = 3.
a(8) = 15 as |8 - a(7)| = |8 - 7| = 1, and 1 + 15 = 16 = 4^2 is the next smallest square. Note a(8) cannot be 3 or 8 as these have previously occurred.

Scott R. Shannon, Image of the first 1000000 terms.

Cf. A349182, A000290, A347595, A347594, A009000, A347754.

A349182 a(0) = 1; for n>0, a(n) is the smallest positive integer such that |n - a(n-1)| + a(n) is a square.

Original entry on oeis.org

1, 1, 3, 1, 1, 5, 3, 5, 1, 1, 7, 5, 2, 5, 7, 1, 1, 9, 7, 4, 9, 4, 7, 9, 1, 1, 11, 9, 6, 2, 8, 2, 6, 9, 11, 1, 1, 13, 11, 8, 4, 12, 6, 12, 4, 8, 11, 13, 1, 1, 15, 13, 10, 6, 1, 10, 3, 10, 1, 6, 10, 13, 15, 1, 1, 17, 15, 12, 8, 3, 14, 7, 16, 7, 14, 3, 8, 12, 15, 17, 1, 1, 19, 17, 14, 10, 5, 18, 11

Offset: 0

Scott R. Shannon, Nov 09 2021

nonn

			a(1) = 1 as |1 - a(0)| = |1 - 1| = 0, and 0 + 1 = 1 = 1^2 is the next smallest square.
a(2) = 3 as |2 - a(1)| = |2 - 1| = 1, and 1 + 3 = 4 = 2^2 is the next smallest square.
a(5) = 5 as |5 - a(4)| = |5 - 1| = 4, and 4 + 5 = 9 = 3^2 is the next smallest square.

Scott R. Shannon, Image for the first 50000 terms.

Cf. A349119, A000290, A347595, A347594, A009000, A347754.

nxt[{n_,a_}]:={n+1,Module[{k=1},While[!IntegerQ[Sqrt[(Abs[n+1-a])+k]],k++];k]}; NestList[ nxt,{0,1},90][[;;,2]] (* Harvey P. Dale, Aug 20 2024 *)

cp's OEIS Frontend

A309500 The minimal number of steps to return to the origin for a self-avoiding walk with step-length n on a 2D plane where at each step the walk must go to an unvisited point with integer coordinates as close as possible to the origin.

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A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

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A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

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A108707 Minimum side in Pythagorean triangles with hypotenuse of n.

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A108708 Maximum side length in Pythagorean triangles with hypotenuse n.

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A136346 Octagonal numbers which are the sums of exactly two positive octagonal numbers.

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A307448 List of pairs of coordinates (x,y) of the visited points for a self-avoiding walk with an incrementing step length confined to one quadrant of a 2D plane where at each step the walk must go to an unvisited point with integer coordinates as close as possible to the origin.

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A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

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A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.

A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2uv, u^2 + v^2), where u and v are positive integers.