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A156253
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Least k such that A054353(k) >= n.
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14
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1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 42, 42, 43, 44, 44, 45, 45, 46, 47, 47, 48, 49, 50
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OFFSET
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1,2
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COMMENTS
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a(n)=1 plus the number of symbol changes in the first n terms of A000002. - Jean-Marc Fedou and Gabriele Fici, Mar 18 2010
Note that Kolakoski's sequence A000002 and Golomb's sequence A001462 have very similar definitions, although the asymptotic behavior of A001462 is well-understood, while that of A000002 is a mystery. The asymptotic behavior of the two hybrids A156253 and A321020 might be worth investigating. (End)
To expand upon N. J. A. Sloane's comments, it's worth noting that Golomb's sequence has a formula from Colin Mallows: g(n) = g(n-g(g(n-1))) + 1, which closely resembles a(n) = a(n-gcd(a(a(n-1)),2)) + 1. - Jon Maiga, May 16 2023
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LINKS
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N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides (Mentions this sequence)
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FORMULA
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Conjecture: a(n) should be asymptotic to 2n/3.
Conjecture: a(n) = (a(a(n-1)) mod 2) + a(n-2) + 1. - Jon Maiga, Dec 09 2021
a(n) = a(n-gcd(a(a(n-1)), 2)) + 1. - Jon Maiga, May 16 2023
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MATHEMATICA
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a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n - 1, 2]}], {n, 3, 80}, {i, 1, a2[[n]]}]; a3 = Accumulate[a2]; a[1] = 1; a[n_] := a[n] = For[k = a[n - 1], True, k++, If[a3[[k]] >= n, Return[k]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 18 2013 *)
a[1] = 1;
a[n_]:=a[n]=a[n-GCD[a[a[n - 1]], 2]]+1
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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