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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 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.)