%I A099884
%S A099884 1,2,3,4,6,5,8,12,10,15,16,24,20,30,17,32,48,40,60,34,51,64,96,80,120,
%T A099884 68,102,85,128,192,160,240,136,204,170,255,256,384,320,480,272,408,340,
%U A099884 510,257,512,768,640,960,544,816,680,1020,514,771,1024,1536,1280,1920
%N A099884 XOR difference triangle of the powers of 2, read by rows.
%C A099884 Define an "XOR difference triangle" for a sequence A by the following 
               process. Start with A in the left-most column. Generate the next 
               column by performing the XOR operation between adjacent terms of 
               the prior column. Repeat this process to generate the XOR difference 
               triangle for A. Further, we define the "XOR BINOMIAL transform" of 
               A as the main diagonal in the XOR difference triangle for A. The 
               XOR BINOMIAL transform is its self-inverse. Let a sequence B be the 
               XOR BINOMIAL transform of A, then we may express B by: B(n) = SumXOR_{k=0..n} 
               A047999(n,k)*A(k), which is equivalent to: B(n) = (C(n,0)mod 2)*A(0) 
               XOR (C(n,1)mod 2)*A(1) XOR (C(n,2)mod 2)*A(2) XOR ... XOR (X(n,n)mod 
               2)*A(n), where the coefficients are C(n,k)(mod 2) = A047999(n,k).
%C A099884 This sequence is a rearrangement of the numbers which are 2^k times distinct 
               Fermat numbers (numbers of the form 2^(2^m) + 1). This matches the 
               sizes of polygons constructible with compass and straightedge (A003401) 
               up to 2^32+1, which is the first nonprime Fermat number. - Frank 
               Adams-Watters (FrankTAW(AT)Netscape.net), Jun 16 2006
%F A099884 T(n, k) = 2^(n-k)*A001317(k). T(n, n) = A001317(n) = SumXOR_{k=0..n} 
               A047999(n, k)*2^k, where SumXOR is the analogue of summation under 
               the binary XOR operation.
%e A099884 The main diagonal equals A001317 (Pascal's triangle mod 2 in decimal):
%e A099884 {1,3,5,15,17,51,85,255,257,771,1285,3855,...},
%e A099884 and defines the XOR BINOMIAL transform of the powers of 2.
%e A099884 Rows begin:
%e A099884 [1],
%e A099884 [2,3],
%e A099884 [4,6,5],
%e A099884 [8,12,10,15],
%e A099884 [16,24,20,30,17],
%e A099884 [32,48,40,60,34,51],
%e A099884 [64,96,80,120,68,102,85],
%e A099884 [128,192,160,240,136,204,170,255],
%e A099884 [256,384,320,480,272,408,340,510,257],
%e A099884 [512,768,640,960,544,816,680,1020,514,771],
%e A099884 [1024,1536,1280,1920,1088,1632,1360,2040,1028,1542,1285],...
%o A099884 (PARI) T(n,k)=local(B);B=0;for(i=0,k,B=bitxor(B,binomial(k,i)%2*2^(n-i)));
               B
%Y A099884 Cf. A047999, A001317.
%Y A099884 Cf. A000215, A003401.
%Y A099884 Sequence in context: A080997 A151942 A054582 this_sequence A118315 A075159 
               A095424
%Y A099884 Adjacent sequences: A099881 A099882 A099883 this_sequence A099885 A099886 
               A099887
%K A099884 nonn,tabl
%O A099884 0,2
%A A099884 Paul D. Hanna (pauldhanna(AT)juno.com), Oct 28 2004