0,2

A100215(n) (ves) = ((-1)^n)*A009116(n+3) (jes) + a(n) (les) + A038503(n+1) (tes) (Sn, below, corresponds to the generating function from above) Coefficients of Sn(z)*(1-z)/(1+z) gives match to A038504 (Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 1") Coefficients of Sn(z)/(1+z) gives match to A038505 (Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 2") Coefficients of Sn(z)/(1-z^2) gives match to A000749 (Number of strings over Z_2 of length n with trace 1 and subtrace 1) The elements 'i, 'j, 'k, i', j', k', 'ii', 'jj', 'kk', 'ij', 'ik', 'ji', 'jk', 'ki', 'kj', e ("floretions") are members of the quaternion product factor space Q x Q /{(1,1), (-1,-1)}. "les" sums over coefficients belonging to basis vectors which squared give the unit e (excluding e itself).

a(n+3) = 4a(n) - 6a(n+1) + 4a(n+2), a(0) = 1, a(1) = 4, a(2) = 9; g.f. (x+1)(x-1)/((2x-1)(2x^2-2x+1)) (a(n)) = lesseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)

Ex. a(2) = 9 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^3 =

1'j + 1'k + 1j' + 1k' + 3'ii' + 2'jj' + 2'kk' + 1'jk' + 1'kj' + 1e

and the sum of the coefficients belonging to basis vectors which squared give the unit e (excluding e itself) is: 3+2+2+1+1 = 9 (see comment).

Floretion Algebra Multiplication Program, FAMP

Cf. A100213, A100214, A100216, A009116, A038503.

Adjacent sequences: A100213 A100214 A100215 this_sequence A100217 A100218 A100219

Sequence in context: A006508 A009875 A027365 this_sequence A138994 A066969 A109593

nonn

Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Nov 11 2004