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 A059906 Index of second half of decomposition of integers into pairs based on A000695. 22

%I

%S 0,0,1,1,0,0,1,1,2,2,3,3,2,2,3,3,0,0,1,1,0,0,1,1,2,2,3,3,2,2,3,3,4,4,

%T 5,5,4,4,5,5,6,6,7,7,6,6,7,7,4,4,5,5,4,4,5,5,6,6,7,7,6,6,7,7,0,0,1,1,

%U 0,0,1,1,2,2,3,3,2,2,3,3,0,0,1,1,0,0,1,1,2,2,3,3,2,2,3,3,4,4,5,5,4,4,5,5,6

%N Index of second half of decomposition of integers into pairs based on A000695.

%C One coordinate of a recursive non-self-intersecting walk on the square lattice Z^2.

%F n = A000695(A059905(n)) + 2*A000695(a(n))

%F To get a(n), write n as Sum b_j*2^j, then a(n) = Sum b_(2j+1)*2^j. - _Vladimir Shevelev_, Nov 13 2008

%F a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=0 and b(k)=A077957(k-1) for k>0. - _Philippe DelĂ©ham_, Oct 18 2011

%F Conjecture: a(n) = n - Sum(k=1..n, sqrt(2)^A007814(k)+(-sqrt(2))^A007814(k))/2 = -Sum(k=1..n, (-1)^k * 2^floor(k/2) * floor(n/2^k)). - _Velin Yanev_, Dec 01 2016

%e A000695(A059905(14)) + 2*A000695(a(14)) = A000695(2) + 2*A000695(3) = 4 + 2*5 = 14.

%e If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(n) = b_1 + b_3*2 = 3. - _Vladimir Shevelev_, Nov 13 2008

%o (Python)

%o def a(n):

%o x=map(int, list(bin(n)[2:]))[::-1]

%o return sum([x[2*i + 1]*2**i for i in xrange(int(len(x)/2))])

%o print [a(n) for n in xrange(105)] # _Indranil Ghosh_, Jun 25 2017

%o (PARI) A059906(n) = { my(t=1,s=0); while(n>0, s += ((n%4)>=2)*t; n \= 4; t *= 2); (s); }; \\ _Antti Karttunen_, Apr 14 2018

%Y Cf. A000695, A057300, A059905.

%K easy,nonn

%O 0,9

%A _Marc LeBrun_, Feb 07 2001

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Last modified August 22 03:39 EDT 2018. Contains 313964 sequences. (Running on oeis4.)