cp's OEIS Frontend

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.

%I A337210 #21 Oct 05 2020 12:50:28
%S A337210 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,
%T A337210 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,
%U A337210 6,12,2,21,1,5,8,9,10,4,18,6,15,1,5,10,10
%N 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.
%C A337210 All approximations are less than or equal to one. An approximation sqrt(n) - sqrt(n-1) < 1 for all n > 1.
%C A337210 Often integers of the form 4n-2 have as their best approximation just the two consecutive integers {n-1, n}.
%C A337210 Those that are not: 20, 21, 25, 27, 30, 31, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 49, 52, ... .
%C A337210 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}.
%F A337210 s = sum(sqrt(i)) for carefully chosen integers i less than n such that s < n yet is the best approximation to n.
%e A337210 For row 1, just the sqrt(0) < sqrt(1);
%e A337210 for row 2, just the sqrt(1) < sqrt(2);
%e A337210 for row 3, just the sqrt(2) < sqrt(3);
%e A337210 for row 4, just the sqrt(3) < sqrt(4);
%e A337210 for row 5, just the sqrt(4) < sqrt(5);
%e A337210 for row 6, sqrt(1) + sqrt(2) < sqrt(6);
%e A337210 for row 7, just the sqrt(6) < sqrt(7);
%e A337210 for row 8, sqrt(1) + sqrt(3) < sqrt(8);
%e A337210 for row 9, just the sqrt(8) < sqrt(9);
%e A337210 for row 10, sqrt(2) + sqrt(3) is the best approximation;
%e A337210 for row 11, sqrt(1) + sqrt(5) < sqrt(11);
%e A337210 for row 12, sqrt(1) + sqrt(6) < sqrt(12);
%e A337210 for row 27, sqrt(1) + sqrt(3) + sqrt(6) is the best approximation;
%e A337210 for row 63, 2*sqrt(3) + 2*sqrt(5) is the best approximation and appears as the integers {12, 20};
%e A337210 for row 107, sqrt(3) + sqrt(6) + sqrt(9) + sqrt(10) is the best approximation;
%e A337210 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};
%e A337210 for row 218, sqrt(1) + sqrt(3) + sqrt(5) + sqrt(6) + sqrt(13) + sqrt(14) is the best approximation; etc.
%e A337210 Triangle begins:
%e A337210 0;
%e A337210 1;
%e A337210 2;
%e A337210 3;
%e A337210 4;
%e A337210 1, 2;
%e A337210 6;
%e A337210 1, 3;
%e A337210 8;
%e A337210 2, 3;
%e A337210 ...
%t A337210 y[x_] := Block[{lst = {x - 1}, min = Sqrt[x] - Sqrt[x - 1], rad = 1, sx = Sqrt[x]},
%t A337210   If[x > 5, lim = (sx - 1)^2;
%t A337210    Do[diff = sx - (Sqrt[a] + Sqrt[b]);
%t A337210     If[diff < min && diff > 0, min = diff; lst = {b, a}; rad = 2],
%t A337210     {a, 2, lim}, {b, 1, a - 1}]];
%t A337210   If[x > 17, lim = (sx - Sum[Sqrt[z], {z, 2}])^2;
%t A337210    Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c]);
%t A337210     If[diff < 0, Continue[]];
%t A337210     If[diff < min && diff > 0, min = diff; lst = {c, b, a}; rad = 3],
%t A337210     {a, 3, lim}, {b, 2, a - 1}, {c, 1, b - 1}]];
%t A337210   If[x > 37, lim = (sx - Sum[Sqrt[z], {z, 3}])^2;
%t A337210    Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d]);
%t A337210     If[diff < 0, Continue[]];
%t A337210     If[diff < min && diff > 0, min = diff; lst = {d, c, b, a};
%t A337210      rad = 4],
%t A337210     {a, 4, lim}, {b, 3, a - 1}, {c, 2, b - 1}, {d, 1, c - 1}]];
%t A337210   If[x > 71, lim = (sx - Sum[Sqrt[z], {z, 4}])^2;
%t A337210    Do[diff = sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e]);
%t A337210     If[diff < 0, Continue[]];
%t A337210     If[diff < min && diff > 0, min = diff; lst = {e, d, c, b, a};
%t A337210      rad = 5],
%t A337210     {a, 5, lim}, {b, 4, a - 1}, {c, 3, b - 1}, {d, 2, c - 1}, {e, 1,
%t A337210      d - 1}]];
%t A337210   If[x > 117, lim = (sx - Sum[Sqrt[z], {z, 5}])^2;
%t A337210    Do[diff =
%t A337210      sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f]);
%t A337210      If[diff < 0, Continue[]];
%t A337210     If[diff < min && diff > 0, min = diff; lst = {f, e, d, c, b, a};
%t A337210      rad = 6],
%t A337210     {a, 6, lim}, {b, 5, a - 1}, {c, 4, b - 1}, {d, 3, c - 1}, {e, 2,
%t A337210      d - 1}, {f, 1, e - 1}]];
%t A337210   If[x > 181, lim = (sx - Sum[Sqrt[z], {z, 6}])^2;
%t A337210    Do[diff =
%t A337210      sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f] +
%t A337210          Sqrt[g]); If[diff < 0, Continue[]];
%t A337210     If[diff < min && diff > 0, min = diff;
%t A337210      lst = {g, f, e, d, c, b, a}; rad = 7],
%t A337210     {a, 7, lim}, {b, 6, a - 1}, {c, 5, b - 1}, {d, 4, c - 1}, {e, 3,
%t A337210      d - 1}, {f, 2, e - 1}, {g, 1, f - 1}]];
%t A337210   If[x > 265, lim = (sx - Sum[Sqrt[z], {z, 7}])^2;
%t A337210    Do[diff =
%t A337210      sx - (Sqrt[a] + Sqrt[b] + Sqrt[c] + Sqrt[d] + Sqrt[e] + Sqrt[f] +
%t A337210          Sqrt[g] + Sqrt[h]); If[diff < 0, Continue[]];
%t A337210     If[diff < min && diff > 0, min = diff;
%t A337210      lst = {g, f, e, d, c, b, a}; rad = 8],
%t A337210     {a, 8, lim}, {b, 7, a - 1}, {c, 6, b - 1}, {d, 5, c - 1}, {e, 4,
%t A337210      d - 1}, {f, 3, e - 1}, {g, 2, f - 1}, {h, 1, g - 1}]];
%t A337210  lst];
%t A337210  Array[ y, 50] // Flatten
%Y A337210 Inspired by A045880.
%Y A337210 Cf. A337211.
%K A337210 nonn,tabf
%O A337210 1,3
%A A337210 _Robert G. Wilson v_, Aug 19 2020