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

1,2

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

Links

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

Example

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

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 *)

Prog

(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.
(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

Crossrefs

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

Cf. A256507.

Sequence in context: A341971 A273824 A137671 * A285715 A328456 A253887

Adjacent sequences: A256943 A256944 A256945 * A256947 A256948 A256949

Keyword

nonn,tabf,nice

Author

Franklin T. Adams-Watters, Apr 19 2015

Status

approved