%I A001317 M2495 N0988
%S A001317 1,3,5,15,17,51,85,255,257,771,1285,3855,4369,13107,21845,65535,
%T A001317 65537,196611,327685,983055,1114129,3342387,5570645,16711935,16843009,
%U A001317 50529027,84215045,252645135,286331153,858993459,1431655765,4294967295,
               4294967297,12884901891,21474836485,64424509455,73014444049,219043332147,
               365072220245,1095216660735,1103806595329,3311419785987
%N A001317 Pascal's triangle mod 2 converted to decimal.
%C A001317 The members are all palindromic in binary, i.e. a subset of A006995. 
               - R. Stephan, Sep 28 2004
%C A001317 a(2n+1) = 3 * a(2n), as follows from a(n)=product_{k in K} (1+2^(2^k)), 
               where K is the set of integers such that n=sum_{k in K} 2^k. -Emmanuel 
               Ferrand, Sep 28 2004
%C A001317 J. H. Conway writes (in Math Forum): at least the first 31 numbers give 
               odd-sided constructible polygons. See also A047999. - M. Dauchez 
               (mdzzdm(AT)yahoo.fr), Sep 19 2005
%C A001317 Decimal number generated by the binary bits of the n-th generation of 
               the Rule 60 elementary cellular automaton. Thus: 1; 0, 1, 1; 0, 0, 
               1, 0, 1; 0, 0, 0, 1, 1, 1, 1; 0, 0, 0, 0, 1, 0, 0, 0, 1; .. - Eric 
               Weisstein (eric(AT)weisstein.com), Apr 08, 2006
%C A001317 One can generate this sequence using simple bitwise operations: a(n) 
               = n XOR ( n << 1 ) where XOR is bitwise XOR and << is bitwise shift 
               left - Joel Madigan (dochoncho(AT)gmail.com), Dec 03 2007
%C A001317 limit(n->inf) log(a(n)-1)/log(3) = n*log(2)/log(3) - Bret Mulvey (bret_x(AT)hotmail.com), 
               May 17 2008
%C A001317 Equals row sums of triangle A166548; e.g., 17 = (2 + 4 + 6 + 4 + 1). 
               [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 16 2009]
%C A001317 Equals row sums of triangle A166555 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 17 2009]
%D A001317 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001317 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001317 H. W. Gould, Exponential Binomial Coefficient Series. Tech. Rep. 4, Math. 
               Dept., West Virginia Univ., Morgantown, WV, Sept. 1961.
%D A001317 R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), 
               no. 1, 3-20.
%D A001317 D. Hewgill, A relationship between Pascal's triangle and Fermat numbers, 
               Fib. Quart., 15 (1977), 183-184.
%D A001317 J.-P. Allouche & J. Shallit, Automatic sequences, Cambridge Univeristy 
               Press, 2003, p 113
%H A001317 T. D. Noe, <a href="b001317.txt">Table of n, a(n) for n=0..300</a>
%H A001317 <a href="Sindx_Ce.html#cell">Index entries for sequences related to cellular 
               automata</a>
%H A001317 Dr. Math, <a href="http://www.mathforum.org/dr.math/faq/formulas/faq/
               regpoly.html">Regular polygon formulas</a>
%H A001317 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rule60.html">Rule 60</a>
%H A001317 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Rule102.html">Rule 102</a>
%F A001317 a(n+1) = a(n) XOR 2a(n), where XOR is binary exclusive OR operator. - 
               Paul D. Hanna (pauldhanna(AT)juno.com), Apr 27 2003
%F A001317 a(n)=prod(e(j, n)=1, 2^(2^j)+1) where e(j, n) is the j-th least significatif 
               digit in binary representation of n (Roberts : see Allouche & Shallit) 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 08 2004
%F A001317 a(2n+1) = 3a(2n). Proof: Since a(n)=product_{k in K} (1+2^(2^k)), where 
               K is the set of integers such that n=sum_{k in K} 2^k, clearly K(2n+1) 
               = K(2n) union {0}, hence a(2n+1)=(1+2^(2^0))*a(2n)=3*a(2n). - Emmanuel 
               Ferrand, Sep 28 2004 - R. Stephan, Sep 28 2004
%F A001317 a(32*n) = 3 ^ (32 * n * log(2) / log(3)) + 1 - Bret Mulvey (bret_x(AT)hotmail.com), 
               May 17 2008
%e A001317 Given a(5)=51, a(6)=85 since a(5) XOR 2a(5) = 51 XOR 102 = 85.
%p A001317 A001317 := proc(n) local k; add((binomial(n,k) mod 2)*2^k, k=0..n); end;
%t A001317 a[n_] := BitXor[ n, BitShiftLeft[ n, 1]] Table[ Nest[a, 1, x], {x, 0, 
               12} ] - Joel Madigan (dochoncho(AT)gmail.com), Dec 03 2007
%o A001317 (PARI) a(n)=sum(i=0,n,(binomial(n,i)%2)*2^i)
%Y A001317 Cf. A000215 (Fermat numbers). Odd-numbered terms give A038183 (1D Cellular 
               Automata Rule 90, "sigma minus")
%Y A001317 Not the same as A053576 nor as A045544.
%Y A001317 Cf. A047999, A054432.
%Y A001317 Sequence in context: A003527 A004729 A045544 this_sequence A053576 A077406 
               A054432
%Y A001317 Adjacent sequences: A001314 A001315 A001316 this_sequence A001318 A001319 
               A001320
%K A001317 nonn,base,easy,nice
%O A001317 0,2
%A A001317 N. J. A. Sloane (njas(AT)research.att.com).