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A012257
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Irregular triangle read by rows: row 0 is {2}; 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.}.
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14
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2, 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
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OFFSET
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0,1
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COMMENTS
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I have sometimes referred to this as Lionel Levine's triangle in lectures. - N. J. A. Sloane, Mar 21 2021
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
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LINKS
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FORMULA
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Sum of row n = A011784(n+2); e.g. row 5 is {1, 1, 1, 2, 2, 3, 4} and the sum of the elements is 1+1+1+2+2+3+4 = 14 = A011784(7). - Benoit Cloitre, Aug 06 2003
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EXAMPLE
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Initial rows are:
{2},
{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},
...
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MAPLE
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T:= proc(n) option remember; `if`(n=0, 2, (h->
seq(i$h[-i], i=1..nops(h)))([T(n-1)]))
end:
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MATHEMATICA
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row[1] = {1, 1}; row[n_] := row[n] = MapIndexed[ Function[ Table[#2 // First, {#1}]], row[n-1] // Reverse] // Flatten; Array[row, 7] // Flatten (* Jean-François Alcover, Feb 10 2015 *)
NestList[Flatten@ MapIndexed[ConstantArray[First@ #2, #1] &, Reverse@ #] &, {1, 1}, 6] // Flatten (* Michael De Vlieger, Jul 12 2017 *)
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PROG
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(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]
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CROSSREFS
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KEYWORD
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AUTHOR
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Lionel Levine (levine(AT)ultranet.com)
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EXTENSIONS
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STATUS
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approved
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