

A256946


Irregular triangle where nth 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
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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 nth's row terms in row n+1.  Reinhard Zumkeller, Apr 22 2015


LINKS

Franklin T. AdamsWatters, 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 (* JeanFrançois 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 !! (n1) !! (k1)
a256946_row n = a256946_tabf !! (n1)
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. AdamsWatters, Apr 19 2015


STATUS

approved



