A135521 - OEIS

%I #19 Dec 18 2019 21:52:07

%S 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 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 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 a(n) = 2^(A091090(n)) - 1.

%H Antti Karttunen, <a href="/A135521/b135521.txt">Table of n, a(n) for n = 1..65537</a>

%F G.f. A(x) satisfies: A(x) = x/(1 - x) + 2*x*A(x^2). - _Ilya Gutkovskiy_, Dec 18 2019

%e From _Omar E. Pol_, Mar 11 2011: (Start)

%e Can be written as a triangle with 2^k entries on each row:

%e 1,

%e 1,3,

%e 1,3,1,7,

%e 1,3,1,7,1,3,1,15,

%e 1,3,1,7,1,3,1,15,1,3,1,7,1,3,1,31,

%e 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 Last term of rows are 2^(k+1) - 1. It appears that the row sums give A001787.

%e (End)

%p GS(2,6,200); [see A135416].

%p # Input n is the number of rows.

%p A135521_list := proc(n) local i,k,NimSum;

%p NimSum := proc(a,b) option remember; local i;

%p zip((x,y)->`if`(x<>y,1,0),convert(a,base,2),convert(b,base,2),0);

%p add(`if`(%[i]=1,2^(i-1),0),i=1..nops(%)) end:

%p seq(seq(NimSum(i,i+1),i=0..2^k-1),k=0..n) end:

%p A135521_list(5); # _Peter Luschny_, May 31 2011

%t Flatten[Table[BitXor[i, i + 1], {k, 0, 10}, {i, 0, -1 + 2^k}]] (* _Peter Luschny_, May 31 2011 *)

%o (PARI)

%o A091090(n) = { my(m=valuation(n+1, 2)); if(n>>m, m+1, max(m, 1)); }; \\ From A091090

%o A135521(n) = ((2^A091090(n))-1); \\ _Antti Karttunen_, Sep 27 2018

%Y Cf. A135416, A091090.

%Y This is Guy Steele's sequence GS(2, 6) (see A135416).

%Y Cf. A000225, A001787. - _Omar E. Pol_, Mar 11 2011

%K nonn,tabf

%O 1,3

%A _N. J. A. Sloane_, based on a message from Guy Steele and _Don Knuth_, Mar 01 2008