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 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. 24
 1, 2, 2, 3, 4, 7, 14, 42, 213, 2837, 175450, 139759600, 6837625106787, 266437144916648607844, 508009471379488821444261986503540, 37745517525533091954736701257541238885239740313139682, 5347426383812697233786139576220450142250373277499130252554080838158299886992660750432 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98). Neil Sloane and Brady Haran, The Levine Sequence, Numberphile video (2021) FORMULA 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). 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). 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 EXAMPLE {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}. MATHEMATICA (* This script is not suitable for computing more than 11 terms *) nmax = 11; ro = {{2, 1}}; a=1; For[n=2, n <= nmax, n++, ro = Transpose[{Table[#[], {#[]}]& /@ 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 *) PROG (Haskell) a011784 = last . a012257_row  -- Reinhard Zumkeller, Aug 11 2014 (R) # This works, as with the others, up to 11. lev2 <- function(x = 10, levprev= NULL){ x <- floor(x) # levlen is the RLE values levterm <-rep(1, x) levlen[] <- 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 CROSSREFS Cf. A012257, A014621, A014622. Sequence in context: A286350 A023105 A281723 * A302487 A032252 A112708 Adjacent sequences:  A011781 A011782 A011783 * A011785 A011786 A011787 KEYWORD nonn,nice AUTHOR Lionel Levine (levine(AT)ultranet.com) 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 STATUS approved

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Last modified October 19 03:10 EDT 2021. Contains 348073 sequences. (Running on oeis4.)