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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A074057 A299756 A163258 * A141063 A138223 A194742

Adjacent sequences:  A337207 A337208 A337209 * A337211 A337212 A337213

KEYWORD

nonn,tabf

AUTHOR

Robert G. Wilson v, Aug 19 2020

STATUS

approved

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Last modified June 17 19:09 EDT 2021. Contains 345085 sequences. (Running on oeis4.)