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A059905 Index of first half of decomposition of integers into pairs based on A000695. 28
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
One coordinate of a recursive non-self-intersecting walk on the square lattice Z^2.
LINKS
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).
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
Sequence in context: A244075 A371459 A354522 * A295301 A308133 A306426
KEYWORD
easy,nonn,look
AUTHOR
Marc LeBrun, Feb 07 2001
STATUS
approved

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Last modified March 28 17:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)