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

1,3

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}.

Links

Table of n, a(n) for n=1..82.

Formula

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

Example

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;
...

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

Crossrefs

Inspired by A045880.

Cf. A337211.

Sequence in context: A299756 A163258 A357143 * A361258 A141063 A138223

Adjacent sequences: A337207 A337208 A337209 * A337211 A337212 A337213

Keyword

nonn,tabf

Author

Robert G. Wilson v, Aug 19 2020

Status

approved