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COMMENT
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a(n) = ves( ('i + 'ii' + 'ij' + 'ik')^n ) a(n) = ves( ('j + 'jj' + 'ji' + 'jk')^n ) a(n) = ves( ('k + 'kk' + 'ki' + 'kj')^n )
The elements x= 'i + 'ii' + 'ij' + 'ik'; y = 'j + 'jj' + 'ji' + 'jk'; and z = 'k + 'kk' + 'ki' + 'kj' are elements of the ring generated from the quaternion factor space Q X Q / {(1,1), (-1,-1)}. Each is represented by a grey shaded area of "Floret's cube". The elements x/2, y/2, z/2 are members of a group, itself a subset of the real algebra generated from Q X Q / {(1,1), (-1,-1)}, which is isomorphic to Q X C_3 (order 24).
This sequence is the term-wise sum of three sequences: a(n) = ves(x^n) = jes(x^n) + les(x^n) + tes(x^n), where jes(x^n)=(1, -6, 8, -24, 16, 0, -64, 384, -512, 1536, -1024, 0, 4096, -24576, 32768, -98304,...), les(x^n)=(3, 0, 0, 0, -48, 0 -192, 0, 0, 0, 3072, 0, 12288, 0, 0, 0,... ), tes(x^n)=(0, 2, 0, -8, 0, -64, 0, -128, 0, 512, 0, 4096, 0, 8192, 0, -32768,...). Concerning "les"- notice that if (..., s, 0, 0, 0, t, ...), then t = -16s and if (..., s, 0, t, ...), then t = 4s
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