1,2

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

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

{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}

Cf. A012257.

Sequence in context: A053638 A051920 A023105 this_sequence A032252 A112708 A032243

Adjacent sequences: A011781 A011782 A011783 this_sequence A011785 A011786 A011787

nonn,nice

Lionel Levine (levine(AT)ultranet.com)

a(12) from C. L. Mallows (colinm(AT)research.avayalabs.com), a(13) from njas, 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 njas, Mar 08 2006