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A012257 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.}.

%I A012257 #50 May 11 2024 21:54:59
%S A012257 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,
%T A012257 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,
%U A012257 9,10,10,11,12,13,14
%N A012257 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.}.
%C A012257 I have sometimes referred to this as Lionel Levine's triangle in lectures. - _N. J. A. Sloane_, Mar 21 2021
%C A012257 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
%H A012257 Reinhard Zumkeller, <a href="/A012257/b012257.txt">Rows n = 0..9 of triangle, flattened</a>
%H A012257 N. J. A. Sloane and Brady Haran, <a href="https://www.youtube.com/watch?v=KNjPPFyEeLo">The Levine Sequence</a>, Numberphile video (2021)
%F A012257 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
%F A012257 T(n,A011784(n+1)) = A011784(n). - _Reinhard Zumkeller_, Aug 11 2014
%e A012257 Initial rows are:
%e A012257 {2},
%e A012257 {1,1},
%e A012257 {1,2},
%e A012257 {1,1,2},
%e A012257 {1,1,2,3},
%e A012257 {1,1,1,2,2,3,4},
%e A012257 {1,1,1,1,2,2,2,3,3,4,4,5,6,7},
%e A012257 ...
%p A012257 T:= proc(n) option remember; `if`(n=0, 2, (h->
%p A012257       seq(i$h[-i], i=1..nops(h)))([T(n-1)]))
%p A012257     end:
%p A012257 seq(T(n), n=0..8);  # _Alois P. Heinz_, Mar 31 2021
%t A012257 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 *)
%t A012257 NestList[Flatten@ MapIndexed[ConstantArray[First@ #2, #1] &, Reverse@ #] &, {1, 1}, 6] // Flatten (* _Michael De Vlieger_, Jul 12 2017 *)
%o A012257 (Haskell)
%o A012257 a012257 n k = a012257_tabf !! (n-1) !! (k-1)
%o A012257 a012257_row n = a012257_tabf !! (n-1)
%o A012257 a012257_tabf = iterate (\row -> concat $
%o A012257                         zipWith replicate (reverse row) [1..]) [1, 1]
%o A012257 -- _Reinhard Zumkeller_, Aug 11 2014, May 30 2012
%Y A012257 Cf. A001462, A011784 (row sums), A012257, A014643, A112798, A181819, A182850-A182858, A296150, A304455.
%K A012257 nonn,tabf,nice,look
%O A012257 0,1
%A A012257 Lionel Levine (levine(AT)ultranet.com)
%E A012257 Initial row {2} added by _N. J. A. Sloane_, Mar 21 2021