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 A049456 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). 27
 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)
 OFFSET 1,4 COMMENTS 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 REFERENCES 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. LINKS 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 FORMULA 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. EXAMPLE 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 ................................. MAPLE 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 MATHEMATICA Flatten[NestList[Riffle[#, Total/@Partition[#, 2, 1]]&, {1, 1}, 10]] (* Harvey P. Dale, Mar 16 2013 *) PROG (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 CROSSREFS 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 KEYWORD nonn,easy,tabf,frac,nice,look AUTHOR STATUS approved

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Last modified December 5 17:04 EST 2022. Contains 358588 sequences. (Running on oeis4.)