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A135521 a(n) = 2^(A091090(n)) - 1.

%I A135521 #19 Dec 18 2019 21:52:07
%S A135521 1,1,3,1,3,1,7,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,
%T A135521 1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,63,1,3,
%U A135521 1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,63,1,3,1,7,1
%N A135521 a(n) = 2^(A091090(n)) - 1.
%H A135521 Antti Karttunen, <a href="/A135521/b135521.txt">Table of n, a(n) for n = 1..65537</a>
%F A135521 G.f. A(x) satisfies: A(x) = x/(1 - x) + 2*x*A(x^2). - _Ilya Gutkovskiy_, Dec 18 2019
%e A135521 From _Omar E. Pol_, Mar 11 2011: (Start)
%e A135521 Can be written as a triangle with 2^k entries on each row:
%e A135521 1,
%e A135521 1,3,
%e A135521 1,3,1,7,
%e A135521 1,3,1,7,1,3,1,15,
%e A135521 1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,
%e A135521 1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,1,3,1,7,1,3,1,15,1,3, 1,7,1,3,1,63,
%e A135521 Last term of rows are 2^(k+1) - 1. It appears that the row sums give A001787.
%e A135521 (End)
%p A135521 GS(2,6,200); [see A135416].
%p A135521 # Input n is the number of rows.
%p A135521 A135521_list := proc(n) local i,k,NimSum;
%p A135521 NimSum := proc(a,b) option remember; local i;
%p A135521 zip((x,y)->`if`(x<>y,1,0),convert(a,base,2),convert(b,base,2),0);
%p A135521 add(`if`(%[i]=1,2^(i-1),0),i=1..nops(%)) end:
%p A135521 seq(seq(NimSum(i,i+1),i=0..2^k-1),k=0..n) end:
%p A135521 A135521_list(5); # _Peter Luschny_, May 31 2011
%t A135521 Flatten[Table[BitXor[i, i + 1], {k, 0, 10}, {i, 0, -1 + 2^k}]] (* _Peter Luschny_, May 31 2011 *)
%o A135521 (PARI)
%o A135521 A091090(n) = { my(m=valuation(n+1, 2)); if(n>>m, m+1, max(m, 1)); }; \\ From A091090
%o A135521 A135521(n) = ((2^A091090(n))-1); \\ _Antti Karttunen_, Sep 27 2018
%Y A135521 Cf. A135416, A091090.
%Y A135521 This is Guy Steele's sequence GS(2, 6) (see A135416).
%Y A135521 Cf. A000225, A001787. - _Omar E. Pol_, Mar 11 2011
%K A135521 nonn,tabf
%O A135521 1,3
%A A135521 _N. J. A. Sloane_, based on a message from Guy Steele and _Don Knuth_, Mar 01 2008