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

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, 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, 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

Offset

0,5

Comments

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

Links

Peter Kagey, 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 (c).

Index entries for sequences related to coordinates of 2D curves

Formula

n = A000695(a(n)) + 2*A000695(A059906(n)).

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

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

G.f.: (1-x)^(-1) * Sum_{j>=0} 2^j*x^(2^j)/(1+x^(2^j)). - Robert Israel, Aug 12 2015

a(n) = A059906(2*n). - Velin Yanev, Dec 01 2016

Example

A000695(a(14)) + 2*A000695(A059906(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_0 + b_2*2 + b_4*2^2 = 5. - Vladimir Shevelev, Nov 13 2008

Maple

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;
map(f, [$0..256]); # Robert Israel, Aug 12 2015

Mathematica

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 *)

Prog

(Ruby)
def a(n)
  (0..n.bit_length/2).to_a.map { |i| (n >> 2 * i & 1) << i}.reduce(:+)
end # Peter Kagey, Aug 12 2015
(Python)
def a(n): return sum([(n>>2*i&1)<<i for i in range(len(bin(n)[2:])//2 + 1)])
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 25 2017, after Ruby code by Peter Kagey
(Python)
def A059905(n): return int(bin(n)[:1:-2][::-1], 2) # Chai Wah Wu, Jun 30 2022
(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

Crossrefs

Cf. A000695, A030308, A059906, A057300, A077957.

Sequence in context: A244075 A371459 A354522 * A295301 A308133 A306426

Adjacent sequences: A059902 A059903 A059904 * A059906 A059907 A059908

Keyword

easy,nonn,look

Author

Marc LeBrun, Feb 07 2001

Status

approved