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%I #34 Dec 29 2022 07:06:48
%S 1,2,4,5,8,9,11,12,16,17,19,20,23,24,26,27,32,33,35,36,39,40,42,43,47,
%T 48,50,51,54,55,57,58,64,65,67,68,71,72,74,75,79,80,82,83,86,87,89,90,
%U 95,96,98,99,102,103,105,106,110,111,113,114,117,118,120,121,128,129
%N a(n) = A005187(n) + 1.
%C Exponent of 2 in the sequences A032184, A052278, A060055, A066318, A088229, A101926.
%C p(n) sequence for k=2, s=0. p(n) = min(j: A046699(j) = n). - _Frank Ruskey_ and Chris Deugau (deugaucj(AT)uvic.ca)
%H C. Deugau and F. Ruskey, <a href="http://www.cs.uvic.ca/~ruskey/Publications/MetaFib/GenMetaFib.html">Complete k-ary Trees and Generalized Meta-Fibonacci Sequences</a>
%H B. Jackson and F. Ruskey, <a href="http://www.cs.uvic.ca/~ruskey/Publications/MetaFib/MetaFib.html">Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes</a>
%H B. Jackson and F. Ruskey, <a href="http://www.combinatorics.org/Volume_13/Abstracts/v13i1r26.html">Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes</a>, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
%H F. Ruskey, C. Deugau, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Ruskey/ruskey6.html">The Combinatorics of Certain k-ary Meta-Fibonacci Sequences</a>, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]
%F Recurrence: a(2n) = 2a(n) + A000120(n) - 1, a(2n+1) = a(2n) + 1.
%F G.f.: (1 / 1-z) * (z + z * sum(z^(2^i) * (s + (1 / (1 - z^(2^k)))),i=0..infinity)). - _Frank Ruskey_ and Chris Deugau (deugaucj(AT)uvic.ca)
%t Table[IntegerExponent[(2 n)!, 2] + 1, {n, 0, 65}] (* or *)
%t Table[2 n - DigitCount[2 n, 2, 1] + 1, {n, 0, 65}] (* _Michael De Vlieger_, Feb 04 2017 *)
%o (PARI) a(n)=1+sum(k=1, n, valuation(k,2)+1)
%o (PARI) a(n)=if(n==0,1,if((n%2)==0,2*a(n/2)+subst(Pol(binary(n)),x,1)-1,a(n-1)+1))
%o (PARI) a(n)=2*n+1-hammingweight(n) \\ _Charles R Greathouse IV_, Dec 29 2022
%o (Python 3.10+)
%o def A101925(n): return (n<<1)-n.bit_count()+1 # _Chai Wah Wu_, Jul 13 2022
%Y Bisection of A089279. First differences are in A001511.
%Y Cf. A000120, A005187, A046699.
%Y Cf. A032184, A052278, A060055, A066318, A088229, A101926.
%K nonn
%O 0,2
%A _Ralf Stephan_, Dec 28 2004
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