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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.}. 13
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 LINKS Reinhard Zumkeller, Rows n = 0..9 of triangle, flattened Neil Sloane and Brady Haran, The Levine Sequence, Numberphile video (2021) FORMULA 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 T(n,A011784(n+1)) = A011784(n). - Reinhard Zumkeller, Aug 11 2014 EXAMPLE 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}, ... MAPLE T:= proc(n) option remember; `if`(n=0, 2, (h->       seq(i\$h[-i], i=1..nops(h)))([T(n-1)]))     end: seq(T(n), n=0..8);  # Alois P. Heinz, Mar 31 2021 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-François 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 (row sums), A012257, A014643, A112798, A181819, A182850-A182858, A296150, A304455. Sequence in context: A293811 A105141 A103961 * A220464 A215975 A071891 Adjacent sequences:  A012254 A012255 A012256 * A012258 A012259 A012260 KEYWORD nonn,tabf,nice,look AUTHOR Lionel Levine (levine(AT)ultranet.com) EXTENSIONS Initial row {2} added by N. J. A. Sloane, Mar 21 2021 STATUS approved

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Last modified January 26 13:09 EST 2022. Contains 350598 sequences. (Running on oeis4.)