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A049456
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Triangle T(n,k) = denominator of fraction in k-th term of n-th row of variant of Farey series. This is also Stern's diatomic array read by rows (version 1).
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26
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1, 1, 1, 2, 1, 1, 3, 2, 3, 1, 1, 4, 3, 5, 2, 5, 3, 4, 1, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,4
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COMMENTS
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Row n has length 2^(n-1) + 1.
A049455/a(n) gives another version of the Stern-Brocot tree.
Define mediant of a/b and c/d to be (a+c)/(b+d). We get A006842/A006843 if we omit terms from n-th row in which denominator exceeds n.
Largest term of n-th row = A000045(n+1), Fibonacci numbers. - Reinhard Zumkeller, Apr 02 2014
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REFERENCES
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J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 154.
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LINKS
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Reinhard Zumkeller, Rows n = 1..13 of table, flattened
C. Giuli and R. Giuli, A primer on Stern's diatomic sequence, Fib. Quart., 17 (1979), 103-108, 246-248 and 318-320 (but beware errors).
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(2) 1929, pp. 59-67.
D. H. Lehmer, On Stern's Diatomic Series, Amer. Math. Monthly 36(1) 1929, pp. 59-67. [Annotated and corrected scanned copy]
M. Shrader-Frechette, Modified Farey sequences and continued fractions, Math. Mag., 54 (1981), 60-63.
N. J. A. Sloane, Stern-Brocot or Farey Tree
Index entries for sequences related to Stern's sequences
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FORMULA
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Each row is obtained by copying the previous row but interpolating the sums of pairs of adjacent terms. E.g. after 1 2 1 we get 1 1+2 2 2+1 1.
Row 1 of Farey tree is 0/1, 1/1. Obtain row n from row n-1 by inserting mediants between each pair of terms.
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EXAMPLE
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0/1, 1/1; 0/1, 1/2, 1/1; 0/1, 1/3, 1/2, 2/3, 1/1; 0/1, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 1/1; 0/1, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, ... = A049455/A049456
Array begins
1...............................1
1...............2...............1
1.......3.......2.......3.......1
1...4...3...5...2...5...3...4...1
1.5.4.7.3.8.5.7.2.7.5.8.3.7.4.5.1
.................................
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MAPLE
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A049456 := proc(n, k)
option remember;
if n =1 then
if k >= 0 and k <=1 then
1;
else
0 ;
end if;
elif type(k, 'even') then
procname(n-1, k/2) ;
else
procname(n-1, (k+1)/2)+procname(n-1, (k-1)/2) ;
end if;
end proc: # R. J. Mathar, Dec 12 2014
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MATHEMATICA
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Flatten[NestList[Riffle[#, Total/@Partition[#, 2, 1]]&, {1, 1}, 10]] (* Harvey P. Dale, Mar 16 2013 *)
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PROG
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(Haskell)
import Data.List (transpose)
a049456 n k = a049456_tabf !! (n-1) !! (k-1)
a049456_row n = a049456_tabf !! (n-1)
a049456_tabf = iterate
(\row -> concat $ transpose [row, zipWith (+) row $ tail row]) [1, 1]
-- Reinhard Zumkeller, Apr 02 2014
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CROSSREFS
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Coincides with A002487 if pairs of adjacent 1's are replaced by single 1's.
Cf. A049455, A007305, A007306, A006842, A006843, A064881-A064886, A070878, A070879.
Cf. A000051 (row lengths), A034472 (row sums), A293160 (distinct terms in each row).
Sequence in context: A132844 A006843 A324797 * A117506 A179205 A055089
Adjacent sequences: A049453 A049454 A049455 * A049457 A049458 A049459
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KEYWORD
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nonn,easy,tabf,frac,nice,look
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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