A011784 Levine's sequence. First construct a triangle as follows. Row 1 is {1,1}; if row n is {r_1, ..., r_k} then row n+1 consists of {r_k 1's, r_{k-1} 2's, r_{k-2} 3's, etc.}; sequence consists of the final elements in each row.

1, 2, 2, 3, 4, 7, 14, 42, 213, 2837, 175450, 139759600, 6837625106787, 266437144916648607844, 508009471379488821444261986503540, 37745517525533091954736701257541238885239740313139682, 5347426383812697233786139576220450142250373277499130252554080838158299886992660750432

Offset: 1

Author

Lionel Levine (levine(AT)ultranet.com)

Keywords

• nonn
• nice

Comments

Additional remarks.

The sequence is generated by this array, the final term in each row forming the sequence:

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

...

where we start with the first row {1 1} and produce the rest of the array recursively as follows:

Suppose line n is {a_1, ..., a_k}; then line n+1 contains a_k 1's, a_{k-1} 2's, etc.

So the fifth line contains three 1's, two 2's, one 3 and one 4.

The sequence is 1,2,2,3,4,7,14,42,213,2837,175450,...,

where the n-th term a(n) is the sum of the elements in row n-2

= the number of elements in row n-1

= the last element in row n

= the number of 1's in row n+1

= ...

If the n-th row is r_{n,i} then

Sum_{i=1..f(n+1)} (a(n+1) - i + 1)*r_{n,i} ) = a(n+3)

Let {a( )} be the sequence; s(i,j) = j-th partial sum of the i-th row,

L(i) is the length of that row and S(i) = its sum. Then

L(i+1) = a(i+2) = S(i) = s(i,a(i+1));

L(i+2) = SUM(s(i,j));

L(i+3) = SUM(s(i,j)*(1+s(i,j))/2) (Allan Wilks).

Eric Rains and Bjorn Poonen have shown (June 1997) that the log of the n-th term is asymptotic to constant times phi^n, where phi = golden number.

This follows from the inequalities S(n) <= a(n)L(n) and S(n+1) >= ([L(n+1)/a(n)]+1) choose 2)*a(n). See N. J. A. Sloane et al., Scans of Notebook pages.

The n-th term is approximately exp(a*phi^n)/I, where phi = golden number, a = .05427 (last digit perhaps 6 or 8), I = .277 (last digit perhaps 6 or 8) (Colin Mallows).

Examples

			{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}.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, Section E25.
• R. K. Guy, What's left?, in The Edge of the Universe: Celebrating Ten Years of Math Horizons, Deanna Haunsperger, Stephen Kennedy (editors), 2006, p. 81.

Links

• Johan Claes, Table of n, a(n) for n = 1..19
• Johnson Ihyeh Agbinya, Computer Board Games of Africa, (2004), see pages 113-114.
• R. K. Guy, What's left?, Math Horizons, Vol. 5, No. 4 (April 1998), pp. 5-7.
• Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^(-1), arXiv:2402.06618 [math.CO], 2024. See pp. 16-17.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• N. J. A. Sloane and Brady Haran, The Levine Sequence, Numberphile video (2021)
• N. J. A. Sloane, Colin Mallows, and Bjorn Poonen, Discussion of A011784. [Scans of pages 150-155 and 164 of my notebook "Lattices 77", from June-July 1997.]

Crossrefs

Cf. A012257, A014621, A014622.

Programs

Haskell

a011784 = last . a012257_row  -- Reinhard Zumkeller, Aug 11 2014

Mathematica

(* This script is not suitable for computing more than 11 terms *) nmax = 11; ro = {{2, 1}}; a[1]=1; For[n=2, n <= nmax, n++, ro = Transpose[{Table[#[[2]], {#[[1]]}]& /@ Reverse[ro] // Flatten, Range[Total[ro[[All, 1]]]]}]; Print["a(", n, ") = ", a[n] = ro // Last // Last]]; Array[a, nmax] (* Jean-François Alcover, Feb 25 2016 *)
NestList[Flatten@ MapIndexed[ConstantArray[First@ #2, #1] &, Reverse@ #] &, {1, 1}, 10][[All, -1]] (* Michael De Vlieger, Jul 12 2017, same limitations as above *)

R

# This works, as with the others, up to 11.
lev2 <- function(x = 10, levprev= NULL){
x <- floor(x[1])
# levlen is the RLE values
levterm <-rep(1,x)
levlen[[1]] <- 2
for ( jl in 2:x) {
rk <- length(levlen[[jl-1]])
for (jrk in 1: rk) {
levlen[[jl]] <- c(levlen[[jl]], rep(jrk, times = levlen[[jl-1]][rk+1-jrk])) }
levterm[jl] <- length(levlen[[jl]]) }
return(invisible(list(levlen=levlen, levterm = levterm) ) ) }
# Carl Witthoft, Apr 01 2021

Formula

a(n+2) = n-th row sum of A012257; e.g., 5th row of A012257 is {1, 1, 1, 2, 2, 3, 4} and the sum of elements is 1+1+1+2+2+3+4=14=a(7) - Benoit Cloitre, Aug 06 2003

a(n) = A012257(n,a(n+1)). - Reinhard Zumkeller, Aug 11 2014

Extensions

a(12) from Colin Mallows, a(13) from N. J. A. Sloane, a(14) and a(15) from Allan Wilks

a(16) from Johan Claes, Jun 09 2004

a(17) (an 85-digit number) from Johan Claes, Jun 18 2004

Edited by N. J. A. Sloane, Mar 08 2006

a(18) (a 137-digit number) from Johan Claes, Aug 19 2008