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A108619 A quaternion-generated sequence calculated using the rules given in the comment box with initial seed x = .5'i + .5'j + .5'k + .5e; version: "base".

%I A108619 #37 Mar 23 2023 06:20:31
%S A108619 1,2,3,2,1,0,-2,-3,0,3,4,3,0,-4,-5,0,5,6,3,-2,-6,-5,2,10,11,2,-7,-8,
%T A108619 -1,8,12,6,-4,-11,-6,5,12,10,0,-11,-10,1,12,14,4,-8,-13,-4,9,14,7,-6,
%U A108619 -14,-9,6,18,15,-2,-18,-17,2,22,23,2,-19,-20,-1,20,24,6,-16,-23,-6,17,24,9,-14,-24,-10,14,27,14,-11,-24,-14,10,27,18,-7
%N A108619 A quaternion-generated sequence calculated using the rules given in the comment box with initial seed x = .5'i + .5'j + .5'k + .5e; version: "base".
%C A108619 Set y = x = .5'i + .5'j + .5'k + .5e Define a(0) = 1 (this is twice the coefficient of 'i in x), then "loop" steps 1-5 as described for A108618. a(n) is given by twice the coefficient of 'i (or 'j or 'k) in y from step 4 inside the n-th loop.
%H A108619 N. J. A. Sloane, <a href="/A108619/b108619.txt">Table of n, a(n) for n = 0..9999</a>
%H A108619 C. Dement, <a href="/A108618/a108618a.jpg">Plot of A108618 against A108619 (patch on)</a>
%H A108619 C. Dement, <a href="/A108618/a108618b.jpg">Plot of A108618 against A108619 (patch off)</a>
%H A108619 C. Dement, <a href="http://fumba.eu/">Floretion Online Multiplier</a> [broken link]
%H A108619 Rémy Sigrist, <a href="/A108619/a108619.png">Colored scatterplot of a(n) for n = 0..9999</a> (where the color is function of n mod 6)
%p A108619 Floretion Algebra Multiplication Program, FAMP Code: 2ibasesum(*)seq[ + .5'i + .5'j + .5'k + .5e]
%t A108619 a[0] = b[0] = 1;
%t A108619 f[n_] := Sign[n]*Mod[n, 2];
%t A108619 a[n_] := a[n] = (1/2)*(a[n-1] - 3*b[n-1]) + 3*f[(1/2)*(a[n-1] + b[n-1])] + f[(1/2)*(a[n-1] - 3*b[n-1])] + 1;
%t A108619 b[n_] := b[n] = (1/2)*(a[n-1] + b[n-1]) + 1;
%t A108619 A108619 = Table[b[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 25 2015, after _Benoit Jubin_ *)
%Y A108619 Cf. A108618, A108620, A272693.
%K A108619 easy,sign,hear,look
%O A108619 0,2
%A A108619 _Creighton Dement_, Jun 22 2005