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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A221919 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

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Last modified December 9 13:50 EST 2019. Contains 329877 sequences. (Running on oeis4.)