|
| |
|
|
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.
|
|
13
| |
|
|
1, 2, 2, 3, 4, 7, 14, 42, 213, 2837, 175450, 139759600, 6837625106787, 266437144916648607844, 508009471379488821444261986503540, 37745517525533091954736701257541238885239740313139682, 5347426383812697233786139576220450142250373277499130252554080838158299886992660750432
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
REFERENCES
| Johnson Ihyeh Agbinya, Computer Board Games of Africa, 2004 (see pages 113-114), http://www.ee.latrobe.edu.au/~johnson/computer%20games/African_Board_Games.pdf
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
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
|
|
|
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 (6/97) 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. 5-th 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 (benoit7848c(AT)orange.fr), Aug 06 2003
|
|
|
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}
|
|
|
CROSSREFS
| Cf. A012257.
Sequence in context: A053638 A051920 A023105 * A032252 A112708 A147558
Adjacent sequences: A011781 A011782 A011783 * A011785 A011786 A011787
|
|
|
KEYWORD
| nonn,nice
|
|
|
AUTHOR
| Lionel Levine (levine(AT)ultranet.com)
|
|
|
EXTENSIONS
| a(12) from C. L. Mallows (colinm(AT)research.avayalabs.com), a(13) from N. J. A. Sloane (njas(AT)research.att.com), a(14) and a(15) from Allan Wilks.
a(16) from Johan Claes (Johan.Claes(AT)luc.ac.be), Jun 09 2004.
a(17) (an 85-digit number) from Johan Claes (Johan.Claes(AT)luc.ac.be), Jun 18 2004.
Edited by N. J. A. Sloane (njas(AT)research.att.com), Mar 08 2006
a(18) (a 137-digit number) from Johan Claes (Johan.Claes(AT)luc.ac.be), Aug 19 2008
|
| |
|
|
