A337210 Irregular triangle read by rows in which row n has the least number of integers such that the sum of the square root of those integers is the best approximation to and less than the square root of n.

0, 1, 2, 3, 4, 1, 2, 6, 1, 3, 8, 2, 3, 1, 5, 1, 6, 12, 3, 4, 2, 6, 3, 5, 2, 7, 4, 5, 1, 11, 1, 12, 2, 10, 5, 6, 1, 14, 3, 10, 1, 3, 5, 6, 7, 1, 3, 6, 2, 15, 2, 3, 5, 7, 8, 1, 3, 8, 2, 4, 5, 4, 14, 8, 9, 6, 12, 2, 21, 1, 5, 8, 9, 10, 4, 18, 6, 15, 1, 5, 10, 10

Offset: 1

Author

Robert G. Wilson v, Aug 19 2020

Keywords

• nonn
• tabf

Comments

All approximations are less than or equal to one. An approximation sqrt(n) - sqrt(n-1) < 1 for all n > 1.

Often integers of the form 4n-2 have as their best approximation just the two consecutive integers {n-1, n}.

Those that are not: 20, 21, 25, 27, 30, 31, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 49, 52, ... .

Sometimes two approximations are equal, i.e.; for n = 39, sqrt(2) + sqrt(4) + sqrt(8) is the same as sqrt(4) + sqrt(18). In this sequence the simplest form is used, i.e.; {4, 18}.

Examples

			For row 1, just the sqrt(0) < sqrt(1);
for row 2, just the sqrt(1) < sqrt(2);
for row 3, just the sqrt(2) < sqrt(3);
for row 4, just the sqrt(3) < sqrt(4);
for row 5, just the sqrt(4) < sqrt(5);
for row 6, sqrt(1) + sqrt(2) < sqrt(6);
for row 7, just the sqrt(6) < sqrt(7);
for row 8, sqrt(1) + sqrt(3) < sqrt(8);
for row 9, just the sqrt(8) < sqrt(9);
for row 10, sqrt(2) + sqrt(3) is the best approximation;
for row 11, sqrt(1) + sqrt(5) < sqrt(11);
for row 12, sqrt(1) + sqrt(6) < sqrt(12);
for row 27, sqrt(1) + sqrt(3) + sqrt(6) is the best approximation;
for row 63, 2*sqrt(3) + 2*sqrt(5) is the best approximation and appears as the integers {12, 20};
for row 107, sqrt(3) + sqrt(6) + sqrt(9) + sqrt(10) is the best approximation;
for row 165, sqrt(1) + 2*sqrt(2) + 2*sqrt(3) + sqrt(5) + sqrt(11) is the best approximation and appears as the integers {1, 5, 8, 11, 12};
for row 218, sqrt(1) + sqrt(3) + sqrt(5) + sqrt(6) + sqrt(13) + sqrt(14) is the best approximation; etc.
Triangle begins:
0;
1;
2;
3;
4;
1, 2;
6;
1, 3;
8;
2, 3;
...
		

Crossrefs

Inspired by A045880.

Cf. A337211.

Programs

Mathematica

y[x_] := Block[{lst = {x - 1}, min = Sqrt[x] - Sqrt[x - 1], rad = 1, sx = Sqrt[x]},
  If[x > 5, lim = (sx - 1)^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b]);
    If[diff < min && diff > 0, min = diff; lst = {b, a}; rad = 2],
    {a, 2, lim}, {b, 1, a - 1}]];
  If[x > 17, lim = (sx - Sum[Sqrt[z], {z, 2}])^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c]);
    If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {c, b, a}; rad = 3],
    {a, 3, lim}, {b, 2, a - 1}, {c, 1, b - 1}]];
  If[x > 37, lim = (sx - Sum[Sqrt[z], {z, 3}])^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d]);
    If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {d, c, b, a};
     rad = 4],
    {a, 4, lim}, {b, 3, a - 1}, {c, 2, b - 1}, {d, 1, c - 1}]];
  If[x > 71, lim = (sx - Sum[Sqrt[z], {z, 4}])^2;
   Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e]);
    If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {e, d, c, b, a};
     rad = 5],
    {a, 5, lim}, {b, 4, a - 1}, {c, 3, b - 1}, {d, 2, c - 1}, {e, 1,
     d - 1}]];
  If[x > 117, lim = (sx - Sum[Sqrt[z], {z, 5}])^2;
   Do[diff =
     sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f]);
     If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff; lst = {f, e, d, c, b, a};
     rad = 6],
    {a, 6, lim}, {b, 5, a - 1}, {c, 4, b - 1}, {d, 3, c - 1}, {e, 2,
     d - 1}, {f, 1, e - 1}]];
  If[x > 181, lim = (sx - Sum[Sqrt[z], {z, 6}])^2;
   Do[diff =
     sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f] +
         Sqrt[g]); If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff;
     lst = {g, f, e, d, c, b, a}; rad = 7],
    {a, 7, lim}, {b, 6, a - 1}, {c, 5, b - 1}, {d, 4, c - 1}, {e, 3,
     d - 1}, {f, 2, e - 1}, {g, 1, f - 1}]];
  If[x > 265, lim = (sx - Sum[Sqrt[z], {z, 7}])^2;
   Do[diff =
     sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f] +
         Sqrt[g] + Sqrt[h]); If[diff < 0, Continue[]];
    If[diff < min && diff > 0, min = diff;
     lst = {g, f, e, d, c, b, a}; rad = 8],
    {a, 8, lim}, {b, 7, a - 1}, {c, 6, b - 1}, {d, 5, c - 1}, {e, 4,
     d - 1}, {f, 3, e - 1}, {g, 2, f - 1}, {h, 1, g - 1}]];
 lst];
 Array[ y, 50] // Flatten

Formula

s = sum(sqrt(i)) for carefully chosen integers i less than n such that s < n yet is the best approximation to n.