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 A088370 Triangle T(n,k), read by rows, where the n-th row is a binary arrangement of the numbers 1 through n. 7
 1, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 5, 3, 2, 4, 1, 5, 3, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 14, 4, 12, 8 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The n-th row differs from the prior row only by the presence of n. See A088371 for the positions in the n-th row that n is inserted. From Clark Kimberling, Aug 02 2007: (Start) At A131966, this sequence is cited as the fractal sequence of the Cantor set C. Recall that C is the set of fractions in [0,1] whose base 3 representation consists solely of 0's and 2's. Arrange these fractions as follows: 0 0, .2 0, .02, .2 0, .02, .2, .22 0, .002, .02, .2, .22, etc. Replace each number x by its order of appearance, counting each distinct predecessor of x only once, getting 1; 1, 2; 1, 3, 2; 1, 3, 2, 4; 1, 5, 3, 2, 4; Concatenate these to get the current sequence, which is a fractal sequence as defined in "Fractal sequences and interspersions". One property of such a sequence is that it properly contains itself as a subsequence (infinitely many times). (End) Row n contains one of A003407(n) non-averaging permutations of [n], i.e., a permutation of [n] without 3-term arithmetic progressions. - Alois P. Heinz, Dec 05 2017 REFERENCES Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168. LINKS Alois P. Heinz, Rows n = 1..141, flattened Eric Weisstein's World of Mathematics, Nonaveraging Sequence Wikipedia, Arithmetic progression FORMULA T(n,n) = 2^(floor(log(n)/log(2))). Construction. The 2n-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n, after multiplying each term by 2. The (2n-1)-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n-1, after multiplying each term by 2. Sum_{k=1..n} k * A088370(n,k) = A309371(n). - Alois P. Heinz, Jul 26 2019 EXAMPLE Row 5 is formed from row 3, {1,3,2} and row 2, {1,2}, like so: {1,5,3, 2,4} = {1*2-1, 3*2-1, 2*2-1} | {1*2, 2*2}. Triangle begins:   1;   1,  2;   1,  3, 2;   1,  3, 2,  4;   1,  5, 3,  2,  4;   1,  5, 3,  2,  6,  4;   1,  5, 3,  7,  2,  6,  4;   1,  5, 3,  7,  2,  6,  4,  8;   1,  9, 5,  3,  7,  2,  6,  4,  8;   1,  9, 5,  3,  7,  2, 10,  6,  4,  8;   1,  9, 5,  3, 11,  7,  2, 10,  6,  4,  8;   1,  9, 5,  3, 11,  7,  2, 10,  6,  4, 12,  8;   1,  9, 5, 13,  3, 11,  7,  2, 10,  6,  4, 12,  8;   1,  9, 5, 13,  3, 11,  7,  2, 10,  6, 14,  4, 12,  8;   1,  9, 5, 13,  3, 11,  7, 15,  2, 10,  6, 14,  4, 12,  8;   1,  9, 5, 13,  3, 11,  7, 15,  2, 10,  6, 14,  4, 12,  8, 16;   1, 17, 9,  5, 13,  3, 11,  7, 15,  2, 10,  6, 14,  4, 12,  8, 16;   ... MAPLE T:= proc(n) option remember;       `if`(n=1, 1, [map(x-> 2*x-1, [T(n-iquo(n, 2))])[],                     map(x-> 2*x,   [T(  iquo(n, 2))])[]][])     end: seq(T(n), n=1..20);  # Alois P. Heinz, Oct 28 2011 MATHEMATICA T[1] = {1}; T[n_] := T[n] = Join[q = Quotient[n, 2]; 2*T[n-q]-1, 2*T[q]]; Table[ T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *) PROG (PARI) {T(n, k) = if(k==0, 1, if(k<=n\2, 2*T(n\2, k) - 1, 2*T((n-1)\2, k-1-n\2) ))} for(n=0, 20, for(k=0, n, print1(T(n, k), ", ")); print("")) CROSSREFS Cf. A003407, A088371, A309371. Diagonal gives A053644. Cf. A049773. - Alois P. Heinz, Oct 28 2011 Sequence in context: A132283 A307081 A256440 * A328719 A113787 A115624 Adjacent sequences:  A088367 A088368 A088369 * A088371 A088372 A088373 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Sep 28 2003 STATUS approved

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Last modified June 24 18:46 EDT 2021. Contains 345419 sequences. (Running on oeis4.)