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A012257 Irregular triangle read by rows: row 1 is {1,1}; if row n is {r_1, ..., r_k} then row n+1 is {r_k 1's, r_{k-1} 2's, r_{k-2} 3's, etc.}. 12
1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 6, 7, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10, 11, 12, 13, 14 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The shape of each row tends to a limit curve when scaled to a fixed size. It is the same limit curve as this continuous version: start with f_0=x over [0,1]; then repeatedly reverse (1-x), integrate from zero (x-x^2/2), scale to 1 (2x-x^2) and invert (1-sqrt(1-x)). For the limit curve we have f'(0) = F(1) = lim A011784(n+2)/(A011784(n+1)*A011784(n)) ~ 0.27887706 (obtained numerically). - Martin Fuller, Aug 07 2006

The row {2} could be safely prepended to this triangle. - Gus Wiseman, May 13 2018

LINKS

Reinhard Zumkeller, Rows n = 1..9 of triangle, flattened

FORMULA

n-th row sum = A011784(n+2); e.g., 5th row is {1, 1, 1, 2, 2, 3, 4} and the sum of elements is 1+1+1+2+2+3+4=14=A011784(7). - Benoit Cloitre, Aug 06 2003

T(n,A011784(n+1)) = A011784(n). - Reinhard Zumkeller, Aug 11 2014

EXAMPLE

First rows are:

{1,1},

{1,2},

{1,1,2},

{1,1,2,3},

{1,1,1,2,2,3,4},

{1,1,1,1,2,2,2,3,3,4,4,5,6,7},

...

MATHEMATICA

row[1] = {1, 1}; row[n_] := row[n] = MapIndexed[ Function[ Table[#2 // First, {#1}]], row[n-1] // Reverse] // Flatten; Array[row, 7] // Flatten (* Jean-Fran├žois Alcover, Feb 10 2015 *)

NestList[Flatten@ MapIndexed[ConstantArray[First@ #2, #1] &, Reverse@ #] &, {1, 1}, 6] // Flatten (* Michael De Vlieger, Jul 12 2017 *)

PROG

(Haskell)

a012257 n k = a012257_tabf !! (n-1) !! (k-1)

a012257_row n = a012257_tabf !! (n-1)

a012257_tabf = iterate (\row -> concat $

                        zipWith replicate (reverse row) [1..]) [1, 1]

-- Reinhard Zumkeller, Aug 11 2014, May 30 2012

CROSSREFS

Cf. A001462, A011784, A012257, A014643, A112798, A181819, A182850-A182858, A296150, A304455.

Sequence in context: A104368 A218878 A240061 * A162320 A241911 A136610

Adjacent sequences:  A012254 A012255 A012256 * A012258 A012259 A012260

KEYWORD

nonn,tabf,nice,look

AUTHOR

Lionel Levine (levine(AT)ultranet.com)

STATUS

approved

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Last modified May 24 17:09 EDT 2019. Contains 323533 sequences. (Running on oeis4.)