n^2=a^2+b^2-1

王守恩

northwolves 发表于 2025-12-22 17:09
{0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 15}

11#可有通项公式？

又：19#的数字串——6, 11, 14, 22, 27, 30, 38, 43, 46, 54, 59, 62, 70, 75, 78, 86, 91, 94, 102, 107, 110, ——\(a^2+b^2-c^4\)  无解的数字串——OEIS有吗？用起来挺厉害的。

更：12#,13#的数字串OEIS也没有！

northwolves

王守恩 发表于 2026-1-2 18:12
11#可有通项公式？

又：19#的数字串——6, 11, 14, 22, 27, 30, 38, 43, 46, 54, 59, 62, 70, 75, 78, 86 ...

$\text{Round}\left[\frac{1}{3} \left(16 n-5 \cos \left(\frac{2 \pi  n}{3}\right)-1\right)\right]$

王守恩

9# northwolves 的公式。谢谢 northwolves！！！

\(2n=(5n+5)^2-(3n+4)^2-(4n+3)^2;2n+1=(5n+3)^2-(3n+2)^2-(4n+2)^2\)

合并了可以这样—— \(n=\big(\frac{9 + (11 + 10 n)\cos(n\pi)}{4 \cos(n\pi)}\big)^2 -\big (\frac{7 + (9 + 6 n)\cos(n\pi)}{4\cos(n\pi)}\big)^2 - \big(\frac{6 + (6 + 8 n) \cos(n\pi)}{4\cos(n\pi)}\big)^2\)

王守恩

有用的资料。

A004437——Numbers that are not the sum of 4 distinct squares.4个不同整数平方和无解的数字。
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 28, 31, 32, 33, 34, 36, 37, 40, 43, 44, 47, 48, 52, 55, 58, 60, 64, 67, 68, 72, 73, 76, 80, 82, 88, 92, 96, 97, 100, 103, 108, 112,

A000534——Numbers that are not the sum of 4 nonzero squares.4个非零整数平方和无解的数字。
1, 2, 3, 5, 6, 8, 9, 11, 14, 17, 24, 29, 32, 41, 56, 96, 128, 224, 384, 512, 896, 1536, 2048, 3584, 6144, 8192, 14336, 24576, 32768, 57344, 98304, 131072, 229376, 393216, 524288, 917504, 1572864, 2097152,

A004441——Numbers that are not the sum of 4 distinct nonzero squares.4个不同非零整数不同平方和无解的数字。
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 52, 53, 55, 56, 58, 59, 60, 61, 64, 67, 68,

4个>1整数平方和无解的数字。{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 32, 34, 35, 39, 41, 44, 46, 51, 55, 56, 62, 67, 89, 96,

4个不同>1整数平方和无解的数字{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,

4个>2整数平方和无解的数字。{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41,

A123069——Odd positive integers that are not the sum of four positive squares.4个奇正整数平方和无解的数字。
1, 3, 5, 9, 11, 17, 29, 41——sm4=Union[Total/@Tuples[Range[10]^2, 4]]; Select[Range[1, 100, 2], !MemberQ[sm4, #]&] (* James C. McMahon, Nov 15 2024 *)

A047701——All positive numbers that are not the sum of 5 nonzero squares.5个>0整数平方和无解的数字。
1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33——list; graph; refs; listen; history; text; internal format)

A004438 Numbers that are not the sum of 5 distinct squares. 不可表示为 5 个不同平方和的数。
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 52, 53, 56, 58, 59, 60, 61, 64, 67, 68, 69, 72, 73, 76, 77, 80, 83,

A123120——Numbers k>0 such that m+k is not the sum of m nonzero squares for any m>5.使得 m(m>5)个非零平方和无解的数字。
1, 2, 4, 5, 7, 10, 13——没看懂。

j[{a_, b_, c_, d_, e_}] := If[0 <= a < b < c < d < e, a^2 + b^2 + c^2 + d^2 + e^2, False]; Union[Complement[Range[500], DeleteCases[Flatten[Map[j, PowersRepresentations[#, 5, 2]] & /@ Range[500]], False]]]