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

1, 2, 4, 8, 12, 24, 30, 64, 72, 120, 130, 288, 300, 420, 434, 1024, 1040, 1296, 1314, 2400, 2420, 2860, 2882, 6912, 6936, 7800, 7826, 11760, 11788, 13020, 13050, 32768, 32800, 35360, 35394, 46656, 46692, 49932, 49970, 96000, 96040, 101640, 101682

Offset: 1

Author

Georgi Guninski, Sep 21 2010

Keywords

• nonn
• uned

Comments

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?

Examples

			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).
		

Links

• Maximilian Hasler, structure & sequences defined by "exotic multiplication", SeqFan Dec 2009. See the whole thread.

Programs

PARI

\\ code by M. F. Hasler |vector(20,i,a(i)[2])|
addi(x,y)=[x[1]+y[1],x[1]*y[2]+x[2]*y[1]];
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)))