A091451 Array T(n,k) read by antidiagonals: (row 0)=squares, (row 1)={numbers m for which the simple continued fraction (CF) of sqrt(m) has period length 1}; once (row n) is defined, let (row n+1) begin with the least positive integer not already in a row and let the rest of (row n+1) be the other m's for which CF(sqrt(m)) has the same period length.
1, 2, 4, 3, 5, 9, 7, 6, 10, 16, 13, 14, 8, 17, 25, 19, 29, 23, 11, 26, 36, 31, 21, 53, 28, 12, 37, 49, 41, 44, 22, 74, 32, 15, 50, 64, 43, 130, 69, 45, 85, 33, 18, 65, 81, 46, 67, 269, 71, 52, 89, 34, 20, 82, 100, 58, 76, 86, 370, 91, 54, 125, 47, 24, 101, 121
Offset: 0
Examples
7 is the least positive integer not in rows 0,1,2, so 7=T(3,0); the period length of sqrt(7) is 4, as is the case with T(3,1)=14, T(3,2)=23, etc. Corner: 1 4 9 16 25 36 49 64 2 5 10 17 26 37 50 65 3 6 8 11 12 15 18 20 7 14 23 28 32 33 34 47 13 29 53 74 85 89 125 173 19 21 22 45 52 54 57 59
Programs
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Mathematica
Map[Map[#[[1]] &, #] &, GatherBy[Map[{#, Flatten[ContinuedFraction[Sqrt[#]]]} &, Range[500]], Length[#[[2]]] &]] (* Peter J. C. Moses, May 11 2023 *)
Extensions
a(47) = T(7,2) corrected by Clark Kimberling, May 20 2023
More terms from Pontus von Brömssen, Nov 23 2024
Comments