A002516 Earliest sequence with a(a(n)) = 2n.
0, 3, 6, 2, 12, 7, 4, 10, 24, 11, 14, 18, 8, 15, 20, 26, 48, 19, 22, 34, 28, 23, 36, 42, 16, 27, 30, 50, 40, 31, 52, 58, 96, 35, 38, 66, 44, 39, 68, 74, 56, 43, 46, 82, 72, 47, 84, 90, 32, 51, 54, 98, 60, 55, 100, 106, 80, 59, 62, 114, 104, 63, 116, 122, 192, 67, 70, 130
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..1000
- Ralf Stephan, Some divide-and-conquer sequences ...
- Ralf Stephan, Table of generating functions
- Index entries for sequences of the a(a(n)) = 2n family
Programs
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Haskell
import Data.List (transpose) a002516 n = a002516_list !! n a002516_list = 0 : concat (transpose [a004767_list, f a002516_list, a017089_list, g $ drop 2 a002516_list]) where f [z] = []; f (_:z:zs) = 2 * z : f zs g [z] = [z]; g (z:_:zs) = 2 * z : g zs -- Reinhard Zumkeller, Jun 08 2015
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Mathematica
a[0] = 0; a[n_ /; Mod[n, 2] == 0] := a[n] = 2*a[n/2]; a[n_ /; Mod[n, 4] == 1] := n+2; a[n_ /; Mod[n, 4] == 3] := 2(n-2); Table[a[n], {n, 0, 67}] (* Jean-François Alcover, Feb 06 2012, after Henry Bottomley *) -
PARI
v2(n)=valuation(n,2) a(n)=2^v2(n)*(-1+3/2*n/2^v2(n)-(-3+1/2*n/2^v2(n))*(-1)^((n/2^v2(n)-1)/2))
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PARI
a(n)=local(t); if(n<1,0,if(n%2==0,2*a(n/2),t=(n-1)/2; 3*t+1/2-(t-5/2)*(-1)^t)) \\ Ralf Stephan, Feb 22 2004
Formula
a(4n) = 2*(a(2n)), a(4n+1) = 4n+3, a(4n+2) = 2*(a(2n+1)), a(4n+3) = 8n+2. - Henry Bottomley, Apr 27 2000
From Ralf Stephan, Feb 22 2004: (Start)
a(2n) = 2*a(n), a(2n+1) = 2n + 3 + (2n - 5)*[n mod 2].
G.f.: Sum_{k>=0} 2^k*t(6t^6 + t^4 + 2t^2 + 3)/(1 - t^4)^2, t = x^2^k. (End)