cp's OEIS Frontend

A059905 Index of first half of decomposition of integers into pairs based on A000695.

%I A059905 #68 Jul 01 2022 09:41:07
%S A059905 0,1,0,1,2,3,2,3,0,1,0,1,2,3,2,3,4,5,4,5,6,7,6,7,4,5,4,5,6,7,6,7,0,1,
%T A059905 0,1,2,3,2,3,0,1,0,1,2,3,2,3,4,5,4,5,6,7,6,7,4,5,4,5,6,7,6,7,8,9,8,9,
%U A059905 10,11,10,11,8,9,8,9,10,11,10,11,12,13,12,13,14,15,14,15,12,13,12,13,14
%N A059905 Index of first half of decomposition of integers into pairs based on A000695.
%C A059905 One coordinate of a recursive non-self-intersecting walk on the square lattice Z^2.
%H A059905 Peter Kagey, <a href="/A059905/b059905.txt">Table of n, a(n) for n = 0..8192</a>
%H A059905 G. M. Morton, <a href="https://dominoweb.draco.res.ibm.com/0dabf9473b9c86d48525779800566a39.html">A Computer Oriented Geodetic Data Base; and a New Technique in File Sequencing</a>, IBM, 1966, with a(n) being section 5.1 step (c).
%H A059905 <a href="/index/Con#coordinates_2D_curves">Index entries for sequences related to coordinates of 2D curves</a>
%F A059905 n = A000695(a(n)) + 2*A000695(A059906(n)).
%F A059905 To get a(n), write n as Sum b_j*2^j, then a(n) = Sum b_(2j)*2^j. - _Vladimir Shevelev_, Nov 13 2008
%F A059905 a(n) = Sum_{k>=0} A030308(n,k)*A077957(k). - _Philippe Deléham_, Oct 18 2011
%F A059905 G.f.: (1-x)^(-1) * Sum_{j>=0} 2^j*x^(2^j)/(1+x^(2^j)). - _Robert Israel_, Aug 12 2015
%F A059905 a(n) = A059906(2*n). - _Velin Yanev_, Dec 01 2016
%e A059905 A000695(a(14)) + 2*A000695(A059906(14)) = A000695(2) + 2*A000695(3) = 4 + 2*5 = 14.
%e A059905 If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(27) = b_0 + b_2*2 + b_4*2^2 = 5. - _Vladimir Shevelev_, Nov 13 2008
%p A059905 f:= proc(n) local L; L:= convert(n,base,2); add(L[2*i+1]*2^i,i=0..floor((nops(L)-1)/2)) end;
%p A059905 map(f, [$0..256]); # _Robert Israel_, Aug 12 2015
%t A059905 a[n_] := Module[{P}, (P = Partition[IntegerDigits[2n, 2]//Reverse, 2][[All, 2]]).(2^(Range[Length[P]]-1))]; Array[a, 100, 0] (* _Jean-François Alcover_, Apr 24 2019 *)
%o A059905 (Ruby)
%o A059905 def a(n)
%o A059905   (0..n.bit_length/2).to_a.map { |i| (n >> 2 * i & 1) << i}.reduce(:+)
%o A059905 end # _Peter Kagey_, Aug 12 2015
%o A059905 (Python)
%o A059905 def a(n): return sum([(n>>2*i&1)<<i for i in range(len(bin(n)[2:])//2 + 1)])
%o A059905 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 25 2017, after Ruby code by _Peter Kagey_
%o A059905 (Python)
%o A059905 def A059905(n): return int(bin(n)[:1:-2][::-1],2) # _Chai Wah Wu_, Jun 30 2022
%o A059905 (PARI) A059905(n) = { my(t=1,s=0); while(n>0, s += (n%2)*t; n \= 4; t *= 2); (s); }; \\ _Antti Karttunen_, Apr 14 2018
%Y A059905 Cf. A000695, A030308, A059906, A057300, A077957.
%K A059905 easy,nonn,look
%O A059905 0,5
%A A059905 _Marc LeBrun_, Feb 07 2001