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A126760
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a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
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33
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0, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 4, 1, 5, 3, 2, 1, 6, 1, 7, 2, 3, 4, 8, 1, 9, 5, 1, 3, 10, 2, 11, 1, 4, 6, 12, 1, 13, 7, 5, 2, 14, 3, 15, 4, 2, 8, 16, 1, 17, 9, 6, 5, 18, 1, 19, 3, 7, 10, 20, 2, 21, 11, 3, 1, 22, 4, 23, 6, 8, 12, 24, 1, 25, 13, 9, 7, 26, 5, 27, 2, 1, 14, 28, 3, 29, 15, 10, 4, 30, 2
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OFFSET
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0,6
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COMMENTS
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For further information see A126759, which provided the original motivation for this sequence.
The odd bisection of the sequence gives A253887, and the even bisection gives the sequence itself.
A254048 gives the sequence obtained when this sequence is restricted to A007494 (numbers congruent to 0 or 2 mod 3).
For all odd numbers k present in square array A135765, a(k) = the column index of k in that array. (End)
A322026 and this sequence (without the initial zero) are ordinal transforms of each other. - Antti Karttunen, Feb 09 2019
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LINKS
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FORMULA
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a(n) = A126759(n)-1. [The original definition.]
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n+5) = 2n + 2.
Or with the last clause represented in another way:
a(0) = 0, a(2n) = a(n), a(3n) = a(n), a(6n+1) = 2n + 1, a(6n-1) = 2n.
Other identities. For all n >= 1:
a(6n-3) = a(4n-2) = a(2n-1) = A253887(n).
(End)
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MATHEMATICA
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f[n_] := Block[{a}, a[0] = 0; a[1] = a[2] = a[3] = 1; a[x_] := Which[EvenQ@ x, a[x/2], Mod[x, 3] == 0, a[x/3], Mod[x, 6] == 1, 2 (x - 1)/6 + 1, Mod[x, 6] == 5, 2 (x - 5)/6 + 2]; Table[a@ i, {i, 0, n}]] (* Michael De Vlieger, Feb 03 2015 *)
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PROG
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(Scheme, with memoizing macro definec)
(definec (A126760 n) (cond ((zero? n) n) ((even? n) (A126760 (/ n 2))) ((zero? (modulo n 3)) (A126760 (/ n 3))) ((= 1 (modulo n 6)) (+ 1 (/ (- n 1) 3))) (else (/ (+ n 1) 3))))
(PARI) A126760(n)={n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2} \\ M. F. Hasler, Jan 19 2016
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CROSSREFS
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Cf. A003586, A003602, A007310, A064989, A249746, A253887, A254048, A273669, A322026 (ordinal transform), A322317, A323881 (Dirichlet inverse), A323882.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Name replaced with an independent recurrence and the old description moved to the Formula section - Antti Karttunen, Jan 28 2015
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STATUS
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approved
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