|
| |
|
|
A088696
|
|
Triangle read by rows, giving number of partial quotients in continued fraction representation of terms in the left branch of the infinite Stern-Brocot tree.
|
|
6
| |
|
|
1, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 5, 4, 3, 4, 5, 4, 3, 4, 3, 2, 1, 2, 3, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 5, 4, 3, 4, 5, 4, 3, 4, 3, 2, 3, 4, 5, 4, 5, 6, 5, 4, 5, 6
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,3
|
|
|
COMMENTS
| A000120 is produced by following each row of this triangle by its reversal.
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 08 2009: (Start)
The row with 8 terms: (1, 2, 3, 2, 3, 4, 3, 2); can be used to generate the
numbers of hydrogen bonds per codon/anti-codon; superimposed on the DNA codon
array of A147995 as follows: top row and left column of an 8X8 array is
composed of the 8 terms (1, 2, 3, 2, 3, 4, 3, 2). If rows and columns have an
offset of "1", then odd rows circulate downward starting from the position (n,n)
Even rows circulate in the opposite direction starting from position (n,n).
This products the array:
1 2 3 2 3 4 3 2
2 1 2 3 4 3 2 3
3 2 1 2 3 2 3 4
2 3 2 1 2 3 2 3
3 4 3 2 1 2 3 2
4 3 2 3 2 1 4 3
3 2 3 4 3 2 1 2
2 3 4 3 2 3 2 1
...
This produces a semi-magic square with a diagonal of (1,1,1...). Using the
simple replacement rule ("complement to 10"): (1->9); (2->8); (3->7); (4->6)
we obtain the chart of DNA hydrogen bonds per codon/anti-codon shown in A147995.
Top row of the hydrogen bond array as well as left column = (9, 8, 7, 8, 7, 6, 7, 8)
Alternatively, using the circulant rule for alternate rows and putting
all 9's along the diagonal, we obtain the chart of hydrogen bonds. (End)
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 30 2009: (Start)
Rows tend to A088748 (which can also be generated from the dragon curve,
A014577) (End)
|
|
|
REFERENCES
| R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pgs. 116-117.
|
|
|
EXAMPLE
| Fractions in the left branch of the infinite Stern-Brocot tree (the fractions between 0 and 1), are:
1/2
1/3 2/3
1/4 2/5 3/5 3/4
1/5 2/7 3/8 3/7 4/7 5/8 5/7 4/5
...
and their corresponding continued fraction representations are:
[2]
[3] [1,2]
[4] [2,2] [1,1,2] [1,3]
[5] [3,2] [2,1,2] [2,3] [1,1,3] [1,1,1,2] [1,2,2] [1,4]
...
with the number of terms in each continued fraction representation generating A088696:
1
1 2
1 2 3 2
1 2 3 2 3 4 3 2
...
|
|
|
MATHEMATICA
| sb[n_List] := Block[{k = l = Length[n], a = n}, While[k > 1, a = Insert[ a, (Numerator[ a[[k]]] + Numerator[ a[[k - 1]]]) / (Denominator[ a[[k]]] + Denominator[ a[[k - 1]]]), k]; k-- ]; a]; sbn[n_] := Complement[ Nest[ sb, {0, 1}, n], Nest[ sb, {0, 1}, n - 1]]; f[n_] := Length /@ (ContinuedFraction /@ sbn[n]) - 1; Flatten[ Table[ f[n], {n, 7}]] (from Robert G. Wilson v Jun 09 2004)
|
|
|
PROG
| (Haskell)
a088696 n = a088696_list !! (n-1)
a088696_list = f [1] where
f (x:xs) = x : f (xs ++ [x + 1 - x `mod` 2, x + x `mod` 2])
-- Reinhard Zumkeller, Mar 07 2011
|
|
|
CROSSREFS
| Cf. A007305, A007306.
A147995 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 08 2009]
A088748, A014577 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 30 2009]
Sequence in context: A192099 A193101 A100661 * A004738 A043554 A005811
Adjacent sequences: A088693 A088694 A088695 * A088697 A088698 A088699
|
|
|
KEYWORD
| nonn,tabf
|
|
|
AUTHOR
| Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 07 2003
|
|
|
EXTENSIONS
| Edited and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 09 2004
|
| |
|
|
