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

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, 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, 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

Offset

0,9

Comments

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

Links

Rémy Sigrist, Table of n, a(n) for n = 0..8192

G. M. Morton, A Computer Oriented Geodetic Data Base; and a New Technique in File Sequencing, IBM, 1966, with a(n) being section 5.1 step (b).

Index entries for sequences related to coordinates of 2D curves

Formula

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

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

a(n) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=0 and b(k)=A077957(k-1) for k>0. - Philippe Deléham, Oct 18 2011

Conjecture: a(n) = n - (1/2)*Sum_{k=1..n} (sqrt(2)^A007814(k) + (-sqrt(2))^A007814(k)) = -Sum_{k=1..n} (-1)^k * 2^floor(k/2) * floor(n/2^k). - Velin Yanev, Dec 01 2016

Example

A000695(A059905(14)) + 2*A000695(a(14)) = A000695(2) + 2*A000695(3) = 4 + 2*5 = 14.
If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(27) = b_1 + b_3*2 = 3. - Vladimir Shevelev, Nov 13 2008

Mathematica

a[n_] := Module[{P}, (P = Partition[IntegerDigits[n, 2]//Reverse, 2][[All, 2]]).(2^(Range[Length[P]]-1))]; Array[a, 105, 0] (* Jean-François Alcover, Apr 24 2019 *)

Prog

(Python)
def a(n):
    x=[int(t) for t in list(bin(n)[2:])[::-1]]
    return sum(x[2*i + 1]*2**i for i in range(int(len(x)//2)))
print([a(n) for n in range(105)]) # Indranil Ghosh, Jun 25 2017
(Python)
def A059906(n): return 0 if n < 2 else int(bin(n)[-2:1:-2][::-1], 2) # Chai Wah Wu, Jun 30 2022
(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

Crossrefs

Cf. A000695, A057300, A059905.

Sequence in context: A234972 A367007 A130326 * A349196 A112046 A076902

Adjacent sequences: A059903 A059904 A059905 * A059907 A059908 A059909

Keyword

easy,look,nonn

Author

Marc LeBrun, Feb 07 2001

Status

approved