A014643 -id:A014643 - cp's OEIS frontend

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.}.

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

Offset: 0

Lionel Levine (levine(AT)ultranet.com)

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

			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},
...

Reinhard Zumkeller, Rows n = 0..9 of triangle, flattened
N. J. A. Sloane and Brady Haran, The Levine Sequence, Numberphile video (2021)

Cf. A001462, A011784 (row sums), A012257, A014643, A112798, A181819, A182850-A182858, A296150, A304455.

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

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

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 *)

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

Initial row {2} added by N. J. A. Sloane, Mar 21 2021

A304679 A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).

Original entry on oeis.org

3, 4, 6, 18, 450, 205439850, 241382525361273331926149714645357743772646450

Offset: 0

Gus Wiseman, May 16 2018

nonn

The first entry 3 is optional so has offset 0.

			The list of multisets with Heinz numbers in the sequence is A014643. The number of k's in row n + 1 is equal to the k-th term of row n. The length of row n is A014644(n).
        3: {2}
        4: {1,1}
        6: {1,2}
       18: {1,2,2}
      450: {1,2,2,3,3}
205439850: {1,2,2,3,3,4,4,4,5,5,5}

Cf. A001462, A014643, A014644, A055932, A056239, A112798, A130091, A181819, A181821, A182850-A182858, A296150, A275870, A304455, A304678.

Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{#[[i]]}],{i,Length[#]}]&,{2},6]

A014644 Form array starting with {1,1}; then i-th term in a row gives number of i's in next row; sequence is formed from final term in each row.

Original entry on oeis.org

1, 2, 2, 3, 5, 11, 38, 272, 6474, 1090483, 4363282578, 2940715000315189, 7930047000157075949085439, 14412592242471457956514645440241289655074, 70636608026754077888330819116433040562582634705380432362008848092

Offset: 1

N. J. A. Sloane

			a(5)=5 because 5 is the last number of the 5th row of A014643, (1,2,2,3,3,4,4,4,5,5,5).

Cf. A014643, A011784.

NestList[Flatten@ MapIndexed[ConstantArray[First[#2], #1] &, #] &, {1, 1}, 8][[All, -1]] (* Michael De Vlieger, Dec 16 2021 *)

log a(n) grows like a constant times phi^n, where phi = golden ratio. - Colin Mallows
a(n) converges to a(n-2)*a(n-1)*phi (within 6 decimals for a(15)). - Johan Claes, Oct 02 2005
Limit_{n -> oo} a(n+2)/(a(n+1)*a(n)) = 1/phi. - Benoit Cloitre, Oct 13 2005

a(1)-a(11) computed by Colin Mallows

a(12)-a(15) computed by Johan Claes Oct 02 2005

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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.}.

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A304679 A prime-multiplicity (or run-length) describing recurrence: a(n+1) = A181821(a(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A014644 Form array starting with {1,1}; then i-th term in a row gives number of i's in next row; sequence is formed from final term in each row.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions