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A184615 Positive parts of the nonadjacent forms for n 6

%I

%S 0,1,2,4,4,5,8,8,8,9,10,16,16,17,16,16,16,17,18,20,20,21,32,32,32,33,

%T 34,32,32,33,32,32,32,33,34,36,36,37,40,40,40,41,42,64,64,65,64,64,64,

%U 65,66,68,68,69,64,64,64,65,66,64,64,65,64,64,64,65,66,68,68,69,72,72,72,73,74,80,80,81,80,80,80,81,82,84,84,85,128

%N Positive parts of the nonadjacent forms for n

%C This sequence together with A184616 (negated negative parts) gives the (signed binary) nonadjacent form (NAF) of n, see fxtbook link.

%C No two adjacent bits in the binary representations of a(n) are 1.

%C No two adjacent bits in the binary representations of a(n)+A184616(n) are 1.

%H Rémy Sigrist, <a href="/A184615/b184615.txt">Table of n, a(n) for n = 0..8192</a>

%H Pages 61-62 of <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>.

%F a(n) - A184616(n) = n

%F a(n) + A184616(n) = A184617(n)

%e The first few nonadjacent forms (NAF) are

%e (dots are used for zeros for better readability):

%e n binary(n) NAF(n)

%e 0: ....... ....... 0 =

%e 1: ......1 ......P 1 = +1

%e 2: .....1. .....P. 2 = +2

%e 3: .....11 ....P.M 3 = +4 -1

%e 4: ....1.. ....P.. 4 = +4

%e 5: ....1.1 ....P.P 5 = +4 +1

%e 6: ....11. ...P.M. 6 = +8 -2

%e 7: ....111 ...P..M 7 = +8 -1

%e 8: ...1... ...P... 8 = +8

%e 9: ...1..1 ...P..P 9 = +8 +1

%e 10: ...1.1. ...P.P. 10 = +8 +2

%e 11: ...1.11 ..P.M.M 11 = +16 -4 -1

%e 12: ...11.. ..P.M.. 12 = +16 -4

%e 13: ...11.1 ..P.M.P 13 = +16 -4 +1

%e 14: ...111. ..P..M. 14 = +16 -2

%e 15: ...1111 ..P...M 15 = +16 -1

%e 16: ..1.... ..P.... 16 = +16

%e 17: ..1...1 ..P...P 17 = +16 +1

%e 18: ..1..1. ..P..P. 18 = +16 +2

%e This sequence gives the words obtained by keeping the 'P' (sum of positive terms in rightmost column), keeping the 'M' gives A184616 (negative sum of negative terms in rightmost column).

%t bin2naf[x_] := Module[{xh, x3, c, np, nm},

%t xh = BitShiftRight[x, 1];

%t x3 = x + xh;

%t c = BitXor[xh, x3];

%t np = BitAnd[x3, c];

%t nm = BitAnd[xh, c];

%t Return[{np, nm}]];

%t a[n_] := bin2naf[n][[1]];

%t Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, May 30 2019, from PARI *)

%o (PARI)

%o bin2naf(x)=

%o { /* Compute (nonadjacent) signed binary representation of x: */

%o local(xh,x3,c,np,nm);

%o xh = x >> 1;

%o x3 = x + xh;

%o c = bitxor(xh, x3);

%o np = bitand(x3, c); /* bits == +1 */

%o nm = bitand(xh, c); /* bits == -1 */

%o return([np,nm]); /* np-nm==x */

%o }

%o { for(n=0,100, v = bin2naf(n); print1(v[1],", "); ); } /* show terms */

%o { for(n=0,100, v = bin2naf(n); print1(v[2],", "); ); } /* terms of A184616 */

%o { for(n=0,100, v = bin2naf(n); print1(v[1]+v[2],", "); ); } /* terms of A184617 */

%o { for(n=0,100, v = bin2naf(n); print1(v[1]-v[2],", "); ); } /* == n */

%Y A184616 (negated negative parts), A184617 (sums of both parts =A184615+A184616).

%K nonn

%O 0,3

%A _Joerg Arndt_, Jan 18 2011

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Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)