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A175841 Fast "exotic addition" a o b = [ a[1]+b[1], a[1]*b[2]+a[2]*b[1] ].

%I A175841 #24 Dec 23 2024 14:53:42
%S A175841 1,2,4,8,12,24,30,64,72,120,130,288,300,420,434,1024,1040,1296,1314,
%T A175841 2400,2420,2860,2882,6912,6936,7800,7826,11760,11788,13020,13050,
%U A175841 32768,32800,35360,35394,46656,46692,49932,49970,96000,96040,101640,101682
%N A175841 Fast "exotic addition" a o b = [ a[1]+b[1], a[1]*b[2]+a[2]*b[1] ].
%C A175841 Define binary operation "o" on pairs of vectors a,b: a o b = [ a[1]+b[1], a[1]*b[2]+a[2]*b[1] ], define scalar multiplication "x" on vector p: 2n x p = (n x p) o (n x p) (2n+1) x p = ((n x p) o (n x p)) o p 1 x p = p. Sequence is: a(n) = (n x p)[2] for p=[1,1] (the first component is n). Other sequences with o associative?
%H A175841 Maximilian Hasler, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2009-December/003299.html">structure & sequences defined by "exotic multiplication"</a>, SeqFan Dec 2009. See the whole thread.
%e A175841 Set p=[1,1], a(2)=o(p,p) [2] = [1+1,1*1+1*1] [2]=[2,2] [2]=2; a(3)=o(o(p,p),p) [2]=o([2,2],[1,1]) [2] =[2+1,2*1+1*2] [2] = [3,4] [2] = 4 (note that computation is fast as in fast exponentiation because of the definition of x).
%o A175841 (PARI) \\ code by _M. F. Hasler_ |vector(20,i,a(i)[2])|
%o A175841 addi(x,y)=[x[1]+y[1],x[1]*y[2]+x[2]*y[1]];
%o A175841 a(n,poi=[1,1])=if(n<=1,poi,if(n%2,n==1&return(poi);addi(a(n-1,poi),poi),poi=a(n\2,poi);addi(poi,poi)))
%K A175841 nonn,uned
%O A175841 1,2
%A A175841 _Georgi Guninski_, Sep 21 2010