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".

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, -1, 8, 12, 6, -4, -11, -6, 5, 12, 10, 0, -11, -10, 1, 12, 14, 4, -8, -13, -4, 9, 14, 7, -6, -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

Offset: 0

Author

Creighton Dement, Jun 22 2005

Keywords

• easy
• sign
• hear
• look

Comments

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.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..9999
• C. Dement, Plot of A108618 against A108619 (patch on)
• C. Dement, Plot of A108618 against A108619 (patch off)
• C. Dement, Floretion Online Multiplier [broken link]
• Rémy Sigrist, Colored scatterplot of a(n) for n = 0..9999 (where the color is function of n mod 6)

Crossrefs

Cf. A108618, A108620, A272693.

Programs

Maple

Floretion Algebra Multiplication Program, FAMP Code: 2ibasesum(*)seq[ + .5'i + .5'j + .5'k + .5e]

Mathematica

a[0] = b[0] = 1;
f[n_] := Sign[n]*Mod[n, 2];
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;
b[n_] := b[n] = (1/2)*(a[n-1] + b[n-1]) + 1;
A108619 = Table[b[n], {n, 0, 100}] (* Jean-François Alcover, Feb 25 2015, after Benoit Jubin *)