A256946 Irregular triangle where n-th row is integers from 1 to n*(n+2), sorted with first squares in order, then remaining numbers by fractional part of the square root.

1, 2, 3, 1, 4, 5, 2, 6, 7, 3, 8, 1, 4, 9, 10, 5, 11, 2, 6, 12, 13, 7, 3, 14, 8, 15, 1, 4, 9, 16, 17, 10, 5, 18, 11, 19, 2, 6, 12, 20, 21, 13, 7, 22, 3, 14, 23, 8, 15, 24, 1, 4, 9, 16, 25, 26, 17, 10, 27, 5, 18, 28, 11, 19, 29, 2, 6, 12, 20, 30, 31, 21, 13, 7, 32, 22, 3, 14, 33, 23, 8, 34, 15, 24, 35

Offset: 1

Author

Franklin T. Adams-Watters, Apr 19 2015

Keywords

• nonn
• tabf
• nice

Comments

This is a fractal sequence.

T(n,k) = T(n+1,A256507(n,k),k), that is, A256507 gives the positions of n-th's row terms in row n+1. - Reinhard Zumkeller, Apr 22 2015

Examples

			The table starts:
  1 2 3
  1 4 5 2 6 7 3 8
  1 4 9 10 5 11 2 6 12 13 7 3 14 8 15
  1 4 9 16 17 10 5 18 11 19 2 6 12 20 21 13 7 22 3 14 23 8 15 24
		

Links

• Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10385 (Rows 1 to 30, flattened.)

Crossrefs

Cf. A005563 (row lengths and last of each row), A083374 (row sums).

Cf. A256507.

Programs

Haskell

import Data.List (sortBy); import Data.Function (on)
a256946 n k = a256946_tabf !! (n-1) !! (k-1)
a256946_row n = a256946_tabf !! (n-1)
a256946_tabf = f 0 [] [] where
   f k us vs = (xs ++ ys) : f (k+1) xs ys where
     xs = us ++ qs
     ys = sortBy (compare `on`
                  snd . properFraction . sqrt . fromIntegral) (vs ++ rs)
     (qs, rs) = span ((== 1) . a010052') [k*(k+2)+1 .. (k+1)*(k+3)]
-- Reinhard Zumkeller, Apr 22 2015

Mathematica

row[n_] := SortBy[Range[n(n+2)], If[IntegerQ[Sqrt[#]], 0, N[FractionalPart[ Sqrt[#]]]]&];
Array[row, 5] // Flatten (* Jean-François Alcover, Sep 17 2019 *)

PARI

arow(n)=vecsort(vector(n*(n+2),k,if(issquare(k),0.,sqrt(k)-floor(sqrt(k)))),,1) \\ This relies on vecsort being stable.