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 A175841 Fast "exotic addition" a o b = [ a[1]+b[1], a[1]*b[2]+a[2]*b[1] ]. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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? LINKS Maximilian Hasler, structure & sequences defined by "exotic multiplication", SeqFan Dec 2009. See the whole thread. EXAMPLE 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). PROG (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))) CROSSREFS Sequence in context: A330022 A032473 A084422 * A293601 A171647 A089821 Adjacent sequences:  A175838 A175839 A175840 * A175842 A175843 A175844 KEYWORD nonn,uned AUTHOR Georgi Guninski, Sep 21 2010 STATUS approved

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Last modified January 17 04:46 EST 2022. Contains 350378 sequences. (Running on oeis4.)